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UNIVERSITY  OF  CALIFORNIA, 

LIBRARY, 

>LOS  ANGELES,  CALIF. 

SUPPLEMENTARY  EDUCATIONAL  MONOGRAPHS 

Publithed  in  coniunMon  with 

THE    SCHOOL   REVIEW   and   THE   ELEMENTARY   SCHOOL  JOURNAL 

Vol.  I  August  27.  1917 

No.  4  Whole  No.  4 


ARITHMETIC  TESTS  AND  STUDIES 

IN  THE  PSYCHOLOGY  OF 

ARITHMETIC 


By 

GEORGE  S.  COUNTS 

Professor  of  Education  in  Delaware  College 


THE  UNIVERSITY  OF  CHICAGO  PRESS 


CHICAGO.  ILLINOIS 


2  78  4     5 


CoFxuGBT  igi7  By 
The  Univxrsity  or  Cbicaoo 


All  Rights  Reserved 


Published  August  rj,  1917 


Composed  and  Printed  By 

The  UDiversity  of  Chicago  Press 

Chicaso,  IllinoU,  U.S.A. 


CLf\ 


TABLE  OF  CONTENTS 

CHAPTEa  PAGE 

I.  Introductory  Statement i 

II.  The  Nature  of  the  Test  and  Collection  of  Data   ....       4 
The  Test 

Series  A  and  B  of  the  Courtis  Tests 
The  Test  Described 

Addition 

Subtraction 

Multiplication 

Division 

Fractions 

Time  Allowance 
Collection  of  Data  ^ 

The  Cleveland  Test 

The  Grand  Rapids  Test 
Summary  Statement 

in.  General  Results 21 

Determination  of  Standards 

Derivation  of  a  System  of  Weights 

The  Use  of  the  Test 

Distributions 

Accuracy 

Summary 

IV.  Types  of  Errors S3 

Addition 

Subtraction 

Multiplication 

Division 

Fractions 

Fractions  of  Like  Denominators 

Fractions  of  Unlike  Denominators 
Summary 


iv  CONTENTS 

CBAPTES  PAGE 

V.  A  Comparison  of  the  Amthmetical  Abilities  of  Certain  Age 
AND  Promotion  Groups 78 

Age  Groups 

Method 

Results 
Promotion  Groups 

Methods 

Resiilta. 
Summary 

VI.  A  Comparison  of  the  Arithmetical  Abilities  of  Certain  Race 

Groups 112 

Method 
Results 
Conclusions 

Vn.  Summary  and  Conclusions 121 

The  Nature  of  the  Test  and  Collection  of  Data 
General  Residts 
Types  of  Errors 

A  Comparison  of  the  Arithmetical  Abilities  of  Certain  Age  and 
Promotion  Groups 

A  Comparison  of  the  Arithmetical  Abilities  of  Certain  Race  Groups 


CHAPTER  I 
INTRODUCTORY  STATEMENT 

This  investigation  is  a  study  of  the  arithmetical  abilities  or 
attainments  of  school  children  as  measured  by  an  arithmetic  test. 
The  study  naturally  falls  into  two  divisions,  the  first  including 
chapters  ii,  iii,  and  iv,  the  second,  chapters  v  and  vi.  In  the 
former,  the  test  used  in  the  investigation  is  described,  and  results 
are  discussed  which  throw  light  on  its  use.  In  the  latter,  two 
special  studies  are  made  in  which  the  test  is  used  as  a  measuring 
instrument.  These  five  chapters  will  now  be  described  in  greater 
detail. 

In  chapter  ii  it  is  shown  that  there  is  a  need  for  a  spiral  test  in 
the  "fundamentals"  of  arithmetic  to  be  used  in  diagnosing  city, 
school,  class,  and  individual  weaknesses  in  the  various  operations 
included  in  the  term  "fundamentals."  It  is  further  pointed  out 
that  Series  A  and  B  of  the  Courtis  standard  tests  are  inadequate 
to  meet  this  need.  The  test  then,  as  developed,  composed  of  15 
sets  of  different  types  of  examples,  is  described  and  analyzed.  This 
is  followed  by  a  statement  concerning  the  collection  of  the  data 
upon  which  the  remainder  of  the  study  is  based. 

The  purpose  of  chapter  iii  is  fivefold:  (i)  In  order  that  the  test 
may  be  of  the  greatest  value  educationally  it  is  necessary  that 
standard  attainments  for  children  in  the  various  grades  in  each  of 
the  15  sets  be  determined.  This  is  done  on  the  basis  of  results 
from  Cleveland  and  Grand  Rapids.  The  validity  of  these  results 
is  discussed  from  the  standpoint  of  the  Courtis  standard  scores. 
(2)  A  system  of  weights  is  derived  by  which  it  is  made  possible  to 
convert  the  scores  made  by  a  particular  group  or  individual  in  the 
15  different  tj^es  of  arithmetical  operations  into  a  single  score  to 
represent  general  arithmetical  attainments  of  the  individual  or 
group.  (3)  The  use  of  the  test  is  discussed  in  detail,  the  method 
by  which  it  may  be  employed  to  diagnose  city,  school,  class,  and 
individual  weaknesses  being  shown.     (4)  Distributions  of  the  scores 


2  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

made  by  groups  of  children  in  the  typical  operations  are  discussed 
for  the  purpose  of  indicating  the  different  types  of  individual 
reaction  to  examples  of  varying  degrees  of  complexity  and  for  the 
purpose  of  pointing  out  certain  differences  in  the  responses  made 
to  the  "fundamentals"  and  to  fractions.  (5)  The  degree  of  accu- 
racy with  which  the  various  types  of  examples  are  worked  is  shown, 
accompanied  by  a  comparison  of  the  curve  of  accuracy  and  the 
curve  of  "rights"  for  one  of  the  sets. 

Chapter  iv  is  a  study  of  errors,  in  which  the  types  of  errors 
made  by  children  in  working  the  different  kinds  of  examples  are 
analyzed.  It  is  of  value  to  the  teacher  to  know  what  sorts  of  errors 
she  may  expect  from  the  pupil  when  the  latter  encounters  the 
different  arithmetical  operations.  The  frequency  of  these  errors  is 
also  determined  in  order  that  the  teacher  may  be  able  to  apply  the 
proper  amount  of  emphasis  at  the  various  points  of  difficulty. 
Because  of  inability  to  isolate  kinds  of  errors  made  in  connection 
with  some  types  of  examples,  since  the  study  was  confined  to  an 
examination  of  records  made  by  pupils,  this  study  is  incomplete. 
It  is  necessary  that  it  be  supplemented  by  experimental  data. 

The  problem  presented  by  the  study  in  chapter  v  is,  in  the  first 
plate,  the  problem  of  measuring  the  attainments  of  various  groups 
of  children  for  the  purpose  of  discovering  differences  in  four  age 
groups  throughout  Grades  3-8  inclusive.  In  the  second  place,  a 
study  is  made  of  certain  promotion  groups  for  the  purpose  of  dis- 
covering differences.  This  division  of  the  study  has  three  parts: 
the  first  relates  to  the  fast  and  slow  pupils  and  is  confined  to  the 
records  of  pupils  in  Grade  8-2;  the  second  is  concerned  with  a 
group  of  pupils  repeating  because  of  failure  to  do  the  work  of  the 
grade,  a  group  repeating  because  of  sickness,  transfer  of  school,  or 
similar  cause,  and  a  group  of  pupils  making  normal  progress,  the 
data  for  this  study  being  secured  from  pupils  in  Grade  7-2  only; 
the  third  has  to  do  with  a  group  of  pupils  in  Grade  8-2  who  had 
failed  below  the  sixth  and  another  group  who  had  failed  above 
the  fifth  grade.  The  differences  found  are  analyzed  and  inter- 
preted. 

In  chapter  vi  a  problem  of  the  same  general  type  as  that  of  the 
previous  chapter  is  encountered.    The  problem  here  is  to  deter- 


INTRODUCTORY  STATEMENT  3 

mine  whether  or  not  there  are  differences  in  arithmetical  attain- 
ments which  follow  racial  lines.  Owing  to  the  meagerness  of  the 
data,  this  study  is  confined  to  five  races,  or  nationalities,  Americans, 
Hollanders,  Germans,  Swedes,  and  Slavs. 

Owing  to  the  fact  that  this  entire  study  has  been  made  on  the 
basis  of  records  made  by  pupils,  it  is  in  many  particulars  incomplete 
and  tentative,  for  there  are  many  matters  that  cannot  be  deter- 
mined by  an  examination  of  records.  Furthermore,  the  conditions 
under  which  the  records  were  made  were  not  suflSiciently  under 
control.  It  is  therefore  evident  that  it  is  necessary  to  supplement 
this  study  by  experimentation. 


CHAPTER  II 
THE  NATURE  OF  THE  TEST  AND  COLLECTION  OF  DATA 

THE  TEST 

In  connection  with  the  Cleveland  Survey  the  demand  arose  for 
an  arithmetic  test  to  measure  the  presence  and  absence  of  arith- 
metical attainments  in  the  school  children  of  that  city.  The  sort 
of  test  desired  was  one  that  would,  on  the  one  hand,  show  the 
general  standing  of  the  city  as  a  whole  in  the  "fundamentals"  of 
arithmetic  and  would,  on  the  other  hand,  be  diagnostic  in  its  char- 
acter, indicating  school,  class,  and  individual  weaknesses  in  each 
of  the  different  types  of  operations  which  enter  into  the  solving  of 
the  more  complex  examples  in  each  of  the  four  fundamental 
operations. 

SERIES  A  AND   B   OF  THE  COURTIS  TESTS 

It  was  felt  by  those  in  charge  of  the  survey  that  no  test  had  as 
yet'been  devised  which  would  exactly  fit  their  needs.  Series  A  of 
the  Courtis  tests  was  unsatisfactory  because,  as  Mr.  Courtis  himself 
has  said,  "the  standards  derived  from  the  use  of  Series  A  .  .  .  . 
are  either  complex  or  of  questionable  value,  owing  to  the  uncer- 
tainty of  their  meaning."*  Tests  Nos.  i,  2,  3,  and  4  of  this  series 
are  merely  tests  of  knowledge  of  the  tables  in  the  four  fundamental 
operations,  and,  since  a  pupil  may  know  his  tables  perfectly  and 
yet  be  quite  unable  to  solve  any  of  the  more  complex  examples, 
and  vice  versa,  these  tests  by  themselves  are  of  little  value.  Test  8, 
the  only  other  test  of  the  fundamentals  in  this  series,  is  of  doubtful 
value.  In  the  first  place,  the  form  in  which  the  examples  appear 
is  not  the  form  to  which  the  child  is  accustomed.  For  example, 
when  called  upon  to  add  two  or  more  numbers,  the  pupil  does  not 
ordinarily  have  them  presented  to  him  in  this  form,  304-7354- 
123=     .    In  order  to  work  the  example  he  must  copy  the  three 

'  S.  A.  Courtis,  Manual  of  Instruction  for  Giving  and  Scoring  the  Courtis  Standard 
Tests,  p.  7. 


THE  NATURE  OF  THE  TEST  AND  COLLECTION  OF  DATA     5 

numbers  in  column  form.  This  consequently  makes  necessary  the 
copying  of  figures  in  the  test,  or  else  the  performing  of  the  operation 
in  a  wholly  unaccustomed  manner.  In  the  second  place,  the  use 
of  the  symbols  introduces  another  factor.  A  pupil  might  be  able 
to  perform  the  required  mathematical  operation  perfectly,  yet  fail 
on  an  example  in  this  test  because  of  unfamiliarity  with  the  symbols. 
If  it  is  desired  to  test  the  knowledge  of  symbols,  a  separate  test 
should  be  devised  for  that  purpose.  In  the  third  place,  a  particular 
score  in  this  test  may  mean  almost  anything  because  of  the  com- 
plex nature  of  the  test.  For  example,  what  may  a  score  of  "four" 
mean?  It  may  mean  either  strength  or  weakness  in  any  one  of 
the  four  operations,  or  it  may  mean  anything  between  these 
extremes. 

Thus,  since  Series  A  is  found  to  be  quite  unsatisfactory,  let  us 
turn  to  Series  B  of  the  Courtis  tests.  The  latter,  when  used  as  a 
supplement  to  the  former,  or  rather  when  substituted  for  Test  8 
of  that  series,  represents  a  distinct  improvement  over  the  earlier 
tests.  The  four  tests  in  Series  B  are  composed  of  four  sets  of  com- 
plex examples  in  the  four  operations.  Test  i  involves  the  addition 
of  columns  of  9  three-place  numbers,  Test  2  the  subtraction  of 
eight-place  numbers  from  eight-  and  nine-place  numbers,  Test  3 
the  multiplication  of  four-  by  two-place  numbers,  and  Test  4  the 
division  of  four-  and  five-  by  two-place  numbers. 

Series  B  supplemented  by  Series  A  is  very  good  so  far  as  it  goes, 
but  it  does  not  go  far  enough.  It  makes  possible  a  measure  of  the 
general  attainment  in  each  of  the  fundamental  operations,  but  does 
nothing  more.  In  a  word,  it  is  not  diagnostic.  For  instance,  sup- 
pose we  have  a  pupil  who  knows  his  addition  tables  perfectly,  as 
indicated  by  a  record  made  in  Test  1  of  Series  A,  but  fails  miser- 
ably on  Test  i  of  Series  B.  These  two  facts  about  the  pupil  are 
worth  knowing,  but  are  of  comparatively  little  value  unless  supple- 
mented by  other  facts.  Why  he  fails  on  the  second  test  is  not 
known.  It  may  be  because  of  failure  to  bridge  the  attention  spans, 
or  of  inability  to  "carry,"  but  the  test  throws  no  light  on  the 
question.  It  is  just  at  this  point  that  Series  A  and  B  of  the  Courtis 
tests  break  down.  It  is  necessary  to  introduce,  between  the  very 
simple  type  of  example  in  the  first  series  and  the  highly  complex 


6  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

type  in  the  second  series,  tests  representing  types  of  intermediate 
complexity.  This  is,  in  fact,  the  logical  evolution  of  Mr.  Courtis' 
own  system  and  is  actually  embodied  in  principle  in  his  Standard 
Practice  Tests. 

THE  TEST  DESCRIBED 

Since  no  existing  test  quite  met  the  needs  of  the  members  of  the 
survey  staff,  they  took  upon  themselves  the  task  of  devising  one. 
In  this  work  the  co-operation  of  Mr.  Courtis  was  secured,  with  the 
result  that  to  him  is  due  whatever  merit  the  test,  as  it  now  stands, 
may  possess. 

In  order  that  the  reader  may  get  a  clear  impression  as  to  the 
nature  of  the  test,  and  that  the  discussion  may  be  the  more  easily 
followed,  the  test  is  here  reproduced  in  full.  Passing  over  for  the 
moment  the  first  page  of  the  test  folder,  since  it  does  not  constitute 
a  part  of  the  test,  the  test  is  seen  to  be  composed  of  15  sets,  desig- 
nated as  Sets  A,  B,  .  .  .  .  O. 

An  examination  of  the  test  shows  it  to  be  composed  of  four  sets 
in  addition  (A,  E,  J,  M),  two  in  subtraction  (B,  F),  three  in  multi- 
plication (C,  G,  L),  four  in  division  (D,  I,  K,  N),  and  two  in  frac- 
tions (H,  O).  Since  the  pupil  begins  with  Set  A  and  takes  each 
set  in  its  proper  order,  the  spiral  character  of  the  test  is  apparent, 
a  feature  which  deserves  some  further  comment.  The  several  sets 
in  each  operation  are  arranged  in  the  test  in  the  order  of  their  com- 
plexity, but  with  them  are  interwoven  the  sets  of  the  other  opera- 
tions. Thus  a  pupil  first  works  on  a  set  of  examples  in  addition, 
then  passes  successively  to  sets  in  subtraction,  multiplication,  and 
division  before  encountering  addition  again.  This  changing  from 
one  type  of  operation  to  another  lessens  the  strain  on  the  pupil 
which  is  involved  in  a  prolonged  test  of  this  sort. 

ADDITION 

As  indicated  above,  there  are  4  sets  in  addition.  Set  A  involves 
the  addition  of  the  simple  combinations.  Set  E  the  addition  of 
colimms  of  5  one-place  numbers.  Set  J  the  addition  of  columns 
of  13  one-place  numbers,  and  Set  M  the  addition  of  columns  of 
5  four-place  numbers.  The  65  examples  of  Set  A  were  taken  from 
Test  I,  Series  A,  of  the  Courtis  tests;  the  16  examples  of  Set  E, 


THE  NATURE  OF  THE  TEST  AND  COLLECTION  OF  DATA     7 

ARITHMETIC  EXERCISES 


Name. 


Age  today. 


Yein         Month! 


Grade. 


School. 


Teacher. 


Date  today. 


Have  you  ever  repeated  the  arithmetic  of  a  grade  because  of  non- 
promotion  or  transfer  from  other  school.    If  so,  name  grade 

Explain  cause 


Inside  this  folder  are  examples  which  you  are  to  work  out  when 
the  teacher  tells  you  to  begin.    "Work  rapidly  and  accurately.    There 
are  more  problems  in  each  set  than  you  can  work  out  in  the  dme  that 
will  be  allowed.    Answers  do  not  count  if  they  are  wroog. 
Begin  and  stop  promptly  at  signals  from  the  teacher. 


A 

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8  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


SET  A— Addition— 

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SET  B— Subtraction— 

9        7        11        8        12        19        13        4        12 
93  61  307  83  6 


8  11        12        5        10        6      11        15     10        12 

09  71  207  89  4 


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5     12        15        5       16       7 
0       9  6         3  8       0 


8        16       9        11 
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Au. 

Ru. 

THE  NATURE  OP  THE  TEST  AND  COLLECTION  OF  DATA     9 


SET  C — Multiplication— 


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SET  D— Division— 

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All. 

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STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


SET  E— Addition 


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SET  F— Subtraction 


616 
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768 
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615 

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1157 
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286 


SET  G— Multiplication— 

2345     9735     8642     6789     2345 
2        5        9        2         6 


2468 
7 


9876 
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At*. 

Rl*. 

THE  NATURE  OF  THE  TEST  AND  COLLECTION  OF  DATA    ii 
SET  H— FracUoiM^ 


S      5 


9       9 


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1  +  1= 
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1  +  1= 
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SET  I— Division— 
4)^5424    7)65982    2)58748    5)41780 


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IS 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


SET  J— Addition 


7 

9 

4 

7 

2 

9 

6 

7 

7 

8 

9 

4 

3 

2 

5 

2 

5 

1 

9 

6 

9 

1 

8 

0 

5 

3 

1 

1 

4 

4 

8 

9 

4 

2 

6 

5 

5 

7 

3 

7 

7 

6 

2 

8 

1 

4 

8 

4 

7 

1 

4 

1 

4 

7 

6 

6 

6 

2 

4 

3 

5 

7 

0 

4 

1 

8 

6 

0 

9 

1 

0 

7 

8 

2 

1 

1 

4 

6 

8 

5 

2 

2 

6 

8 

5 

5 

5 

8 

5 

3 

3 

5 

2 

1 

3 

9 

3 

6 

1 

3 

1 

5 

2 

9 

7 

3 

1 

3 

9 

5 

4 

9 

8 

6 

3 

2 

4 

2 

1 

3 

3 

7 

2 

6 

5 

7 

3 

1 

9 

7 

3 

3 

6 

7 

9 

4 

2 

3 

4 

5 

2 

4 

6 

7 

6 

8 

0 

6 

8 

9 

8 

4 

2 

2 

9 

8 

3 

1 

7 

5 

6 

1 

4 

4 

5 

8 

9 

2 

9 

8 

5 

9 

6 

5 

6 

7 

5 

4 

6 

8 

9 

4 

SET  K— Division— 


21J773      52)1768    417779        2274?2        3lj837 


4^^966      237783       72)1656     81)972        73)1^7^ 


21J294      62)1984    31)527        52)2184      41)98^ 


327T?3      51)2397     82)1968     71)3692^      227784 


41)1681     337693      61)1586       53^ll66      3l74?6 


At*.' 

Rta. 

THE  NATURE  OF  THE  TEST  AND  COLLECTION  OF  DATA    13 


SET  L — Multiplication — 


8246 
29 


3597 
73 


5739 

85 


7593 
64 


6428 
58 


SET  M— Addition— 


SET  N— Division— 
67)32763       48)28464 


2648 
46 


8563 
207 


7493 

8937 

8625 

2123 

5142 

3691 

9016 

6345 

4091 

1679 

0376 

4526 

6487 

2783 

3844 

5555 

4955 

7479 

7591 

4883 

8697 

6331 

9314 

2087 

6166 

1341 
9149 

7314 
6268 

6808 
9397 

5507 
7337 

8165 

5226 

8243 

2883 

8467 

7725 

6158 

2674 

6429 

2584 

0251 

8331 

3732 

9669 

9298 

0058 

7535 

5493 

4641 

5114 

7404 

2398 

5223 

3918 

7919 

8154 

2575 

97>36684       59)29382 


78)69888        88^^44^^       69)40296       38)26562 


Ate. 

RU. 

14  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

SET  0~Fraclion» — 


15  6 

1  1- 
14   4 

3   5 
T^7= 

1  L- 

6  21" 

5  19 

6  ^20~ 

11  1^ 

12  8  ~ 

1x1= 
6   10 

i.  11- 
6   15 

12   8 

20  J__ 

21  6 

4   18 

3 Z^_ 

%      10 

AU. 

Rl.. 

Instructions  for  Examiners 

Have  the  children  fill  out  the  blanks  at  the  top  of  the  first  page. 
Have  them  start  and  stop  work  together.  Let  there  be  an  interval  of 
h^f  a  minute  between  each  set  of  exampfes.  Take  two  days  for  the 
test;  down  through  I  the  first  day,  and  complete  the  test  on  the  next 
day.    The  time  allowances  given  below  must  be  followed  exactly. 


Set  A. 30  seconds 

Set  B.., 30  seconds 

Set  C -.30  seconds 

Set  D 30  seconds 

Set  E 30  seconds 


Set  R. 


1  minute 


Set  G. 1  minute 

Set  H 30  seconds 

Set   I I  Qiinute 

■Set  J„ 2  minutes 


Set  K 2  minutes 

Set  L 3  minutes 

Set  M 3  minutes 

Set  N 3  minutes 

Set  O 3  minutes 


Have  the  children  exchange  papers.  Read  the  answers  aloud 
and  let  the  children  mark  each  example  that  is  correct,  "C."  For 
each  set  let  them  count  the  number  of  problems  attempted  and  the 
number  of  C's  and  write  the  numbers  in  the  appropriate  columns  at  the 
right  of  the  page. 


The  records  should  then  be  transcribed  to  the  first  page, 
verify  the  results  set  down  by  the  pupils. 


Please 


THE  NATURE  OF  THE  TEST  AND  COLLECTION  OF  DATA     15 

the  14  examples  of  Set  J,  and  the  12  examples  of  Set  M  were  taken 
wholly  or  in  part  from  Lessons  4,  23,  and  27,  respectively,  of  the 
Courtis  Standard  Practice  Tests. 

These  4  types  of  examples  in  addition  were  chosen  because  the 
solution  of  the  examples  in  each  succeeding  set  involves  a  mental 
process  not  present  in  the  immediately  preceding  set,  which  marks 
it  ofif  as  a  type.  Thus  Set  A  represents  the  very  simplest  sort  of 
addition,  the  combining  of  2  one-place  numbers.  In  Set  E  the 
pupil  must  not  only  combine  two  numbers,  but  must  hold  this  sum 
in  his  mind  and  combine  it  in  turn  with  a  third  number,  and  so 
on  through  four  combinations.  At  first  glance  Set  J  seems  to  be 
of  the  same  type  as  Set  E,  the  difference  being  merely  one  of  quan- 
tity, but  such  is  not  the  case.  Twelve  combinations  must  be  made 
instead  of  four.  Now  the  span  of  attention  has  limits.  Anyone 
who  has  ever  attempted  to  add  a  long  column  of  figures  knows 
what  this  means.  The  addition  of  one  figure  after  another  from 
the  first  figure  in  the  column  to  the  last  is  not  one  continuous 
process,  but  is  broken  up  into  segments.  That  is,  the  individual 
adds  up  to  a  certain  point,  holds  the  sum  in  his  mind  as  the  atten- 
tion wavers,  and  then  continues  the  addition  of  the  column  as  the 
attention  returns.  This  is  called  "bridging  the  attention  spans" 
and  is  a  mental  process  called  forth  in  the  addition  of  the  long 
columns  in  Set  J.  There  is  one  other  operation  that  the  pupil  must 
learn  to  perform  successfully  before  he  can  become  a  competent 
adder,  and  that  is  "carrying."  For  testing  ability  to  perform  this 
operation  Set  M  appears  in  the  test.  In  the  addition  of  these 
columns  the  pupil  must  "carry"  a  result  forward  from  the  addition 
of  one  column  to  the  next.  Thus  the  4  sets  in  addition  indicate 
ability  or  lack  of  ability  (i)  in  performing  the  simple  addition  com- 
binations, (2)  in  adding  a  third  number  to  a  sum  secured  by  the 
addition  of  two  numbers,.  (3)  in  bridging  the  attention  spans,  and 
(4)  in  "carrying." 

SUBTRACTION 

There  are  but  2  sets  in  subtraction  in  the  test.  The  first.  Set  B, 
is  made  up  of  the  simple  combinations;  and  the  second.  Set  F, 
involves  the  subtraction  of  three-place  numbers  from  three-  and 


l6  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

four-place  numbers.  The  examples  in  the  former  set  were  taken 
from  Test  2,  Series  A,  of  the  Courtis  tests,  and  those  in  the  latter 
from  Lesson  20  of  the  Courtis  Standard  Practice  Tests. 

Subtraction  is  confined  to  2  sets  because,  for  diagnostic  pur- 
poses, they  are  sufficient.  The  only  operation  that  is  added  in  the 
more  complex  forms  of  subtraction,  not  found  in  the  simple  com- 
binations, is  that  of  borrowing.  This  is  demanded  in  the  examples 
of  Set  F  just  as  much  as  in  the  larger  examples. 

MULTIPLICATION 

Multiplication  appears  in  3  sets.  Set  C  involves  the  simple 
combinations,  Set  G  the  multiplication  of  four-place  by  one-place 
numbers,  and  Set  L  the  multiplication  of  four-place  by  two-place 
numbers.  The  50  examples  in  Set  C  were  taken  from  Test  3, 
Series  A,  of  the  Courtis  tests;  the  20  examples  in  Set  G  were 
specially  devised  under  the  supervision  of  Mr.  Courtis  for  this 
test;  and  the  8  examples  in  Set  L  were  taken  from  Test  3,  Series  B, 
of  the  Courtis  tests. 

The  first  set  tests  knowledge  of  the  tables.  In  the  second  the 
pupil  must  "carry"  results  forward.  And  in  the  third,  Set  L,  the 
operation  is  further  complicated  by  the  demand  for  knowledge  of 
the  mechanics  of  handling  the  product  of  the  multiplication  and  the 
second  term  of  the  multiplier.  The  addition  of  the  partial  products 
is  also  introduced. 

DIVISION 

Four  sets  are  given  over  to  division,  D,  I,  K,  and  N.  The 
simple  combinations  appear  in  Set  D,  the  division  of  five-place 
by  one-place  numbers  in  Set  I,  the  division  of  three-  and  four-place 
numbers  by  two-place  numbers  in  Set  K,  and  the  division  of  five- 
place  numbers  by  two-place  numbers  in  Set  N.  The  49  examples 
in  the  first  set  were  taken  from  Test  4,  Series  A,  of  the  Courtis  tests; 
the  12  examples  in  Set  I  were  taken  from  Lesson  31  of  the  Courtis 
Standard  Practice  Tests;  and  the  other  two  sets,  K  and  N,  made 
up  of  25  and  8  examples,  respectively,  were  specially  devised  for 
this  test. 

As  in  the  sets  for  the  other  three  operations,  the  attempt  was 
here  made  to  introduce  into  the  test  examples  embodying  the 


THE  NATURE  OF  THE  TEST  AND  COLLECTION  OF  DATA     17 

different  types  of  difficulty  that  are  encountered  in  division.  Set  D 
tests  knowledge  of  the  tables.  Set  I  is  made  up  of  more  complex 
examples  in  short  division  which  differ  from  the  examples  in  Set  D 
by  the  introduction  of  the  operation  of  carrying.  Sets  K  and  N 
are  sets  in  long  division.  The  former  represents  the  very  simplest 
type  of  this  operation,  since  there  is  no  carrying  required  in  the 
multiplication  and  no  borrowing  in  the  subtraction.  The  latter, 
on  the  other  hand,  is  much  more  complex,  involving  both  carrying 
and  borrowing. 

FRACTIONS 

For  the  purpose  of  testing  the  ability  of  pupils  to  apply  the  four 
fundamental  operations  to  the  working  of  fractions  2  sets  of  fractions 
were  placed  in  the  test,  Set  H  and  Set  O.  Both  sets,  the  one  made 
up  of  24  examples  and  the  other  of  12,  were  specially  devised  for 
this  test. 

The  examples  in  Set  H  are  very  simple,  involving  the  addition 
and  subtraction  of  fractions  of  like  denominators.  In  Set  O  frac- 
tions of  unlike  denominators  are  to  be  added,  subtracted,  multiplied, 
and  divided.  These  sets  of  fractions,  it  will  be  noted,  differ  from 
the  other  sets  in  that  they  are  not  homogeneous.  In  the  first  there 
are  two  different  types  of  operations  to  be  performed,  and  in  the 
second  there  are  four.  This  is  freely  acknowledged  as  a  defect. 
But,  since  the  test  was  to  be  used  in  the  survey,  it  was  necessary 
that  its  scope  be  limited;  and,  since  the  testing  of  attainments  in 
fractions  was  felt  to  be  more  or  less  experimental,  it  was  thought 
that  the  fractions  should  be  sacrificed  rather  than  the  fundamental 
operations. 

TIME  ALLOWANCE 

A  word  should  be  said  about  the  time  allowances  given  to  the 
several  sets.  The  child  is  not  allowed  to  begin  with  the  first  set 
and  to  work  an  indefinite  time  on  it  or  any  following  set.  On  the 
contrary,  as  indicated  by  the  time  allowances  given  on  the  last 
page  of  the  test,  the  pupil  is  allowed  to  work  a  specified  time  on 
each  set.  This  time  ranges  from  30  seconds  for  the  easier  sets  to 
3  minutes  for  the  more  difficult  sets.  In  each  case  the  attempt 
was  made  to  make  the  time  allowance  large  enough  to  enable  even 


i8  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

the  slowest  pupil  to  work  at  least  one  of  the  examples,  and  yet 
small  enough  to  prevent  even  the  most  rapid  pupil  from  exhausting 
the  possibilities  of  the  set.  Thus  the  test  is  a  speed  test  with  a 
definite  time  allowance  given  to  each  of  the  15  sets. 

COLLECTION  OF  DATA 

The  data  on  which  the  present  study  is  based  were  secured  from 
two  sources,  viz.,  Cleveland  and  Grand  Rapids.  Since  there  were 
slight  differences  in  the  tests  themselves  and  in  the  giving  of  the 
tests  in  the  two  cities,  each  will  be  treated  separately. 

THE  CLEVELAND  TEST 

The  test  as  given  to  the  children  of  the  Cleveland  schools  was 
slightly  different  from  that  just  described.  In  the  Cleveland  test 
the  result  "21"  was  repeated  so  frequently  in  Set  K  that  some  of 
the  pupils  taking  the  test,  after  working  several  examples  of  the 
set  and  finding  the  answers  to  be  "21"  in  almost  every  instance, 
wrote  down  "  21 "  as  the  answer  to  the  remaining  examples  without 
actually  working  them.  In  the  light  of  this  experience  Set  K  was 
modified  so  as  to  avoid  the  repetition  of  this  result.  Set  L  was 
modified  by  giving  more  space  for  working  the  examples,  because 
the  Cleveland  results  showed  that  insufficient  space  had  been 
given.  Set  0  was  also  modified.  In  its  earlier  form  the  examples 
in  the  addition  of  fractions  constituted  one  column,  those  in  sub- 
traction another,  those  in  multiplication  another,  and  those  in 
division  another.  When  they  appeared  in  this  form  it  was  found 
that  quite  frequently  a  pupil  would  select  the  examples  in  multi- 
plication and  avoid  the  more  difficult  examples  of  the  other  opera- 
tions. To  place  a  check  on  this  tendency  the  four  types  of  examples 
were  intermingled,  as  seen  in  the  test  in  its  present  form. 

The  tests  were  given  on  June  4,  7,  and  8,  1915,  to  the  B  sections 
of  Grades  3—8  inclusive.  The  teachers  gave  the  tests,  following  the 
instructions  given  on  the  test  sheet  and  certain  other  instructions 
sent  out  to  the  principals  of  the  schools  from  the  office  of  the  super- 
intendent.* The  scoring  was  done  by  the  pupils  under  the  super- 
vision of  the  teacher. 

•  Charles  H.  Judd,  Measuring  the  Work  of  the  Public  Schools,  p.  245. 


THE  NATURE  OF  THE  TEST  AND  COLLECTION  OF  DATA     19 

THE   GRAND   RAPIDS  TEST 

The  test  as  described  in  this  chapter  was  given  to  both  sections 
of  Grades  3-8  inclusive  on  February  28  and  29  and  March  i,  2, 
and  3,  1 91 6.  A  great  deal  more  care  was  taken  here  than  in  Cleve- 
land to  insure  the  results  against  error.  In  the  first  place,  the 
writer  was  present  at  a  meeting  of  the  principals  from  all  of  the 
schools,  where  the  test  was  carefully  gone  over  and  the  method  of 
giving  the  test  explained.  In  the  second  place,  the  request  was 
made  that  one  person,  preferably  the  principal,  do  all  the  timing 
in  each  school,  and  that  the  testing  be  begun  in  the  lower  grades 
and  proceed  upward,  so  that  the  examiners  might  be  somewhat 
experienced  in  the  giving  of  the  test  when  the  more  important 
grades  were  tested — more  important  because  it  is  only  in  the  upper 
three  grades  that  the  children  are  able  to  work  examples  in  all  the 
sets.  In  the  third  place,  the  teachers  and  the  pupils  in  the  Grand 
Rapids  schools  were  familiar  with  the  Courtis  practice  tests.  The 
teachers  were  consequently  to  some  degree  experienced  examiners, 
and  the  children  were  acquainted  with  the  signals  for  beginning 
and  stopping  work.  In  the  fourth  place,  the  writer  personally  con- 
ducted the  tests  in  50  classes  in  8  schools. 

TABLE  I 

Number  of  Classes  Tested 


Grade 

Cleveland 

Grand  Rapids 

Total 

2 

8S 
87 
90 

87 
86 

8S 

64 
62 
58 
S3 
46 
31 

149 
149 
148 

4 

c 

6 

140 

7 

132 
116 

8 

Total 

520 

314 

834 

From  these  two  sources,  as  shown  in  Table  I,  results  were 
secured  from  834  classes,  520  in  the  Cleveland  schools  and  314  in 
the  schools  of  Grand  Rapids.  The  number  of  classes  is  given  rather 
than  the  number  of  children  tested  because  the  medians  in  the 
general  tables  to  be  discussed  in  the  following  chapter  are  medians 
of  class  standings  and  not  of  individual  standings. 


20  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

SUMMARY  STATEMENT 

In  summary,  the  present  test  is  a  speed  test  which  measures 
attainments  and  indicates  weaknesses  in  the  four  fundamental 
operations  and  fractions.  In  addition  it  tests  knowledge  of  tables, 
the  ability  to  add  short  columns,  to  bridge  the -attention  spans,  and 
to  "carry";  in  subtraction  it  tests  knowledge  of  the  tables  and  the 
ability  to  "borrow";  in  multiplication  it  tests  knowledge  of  the 
tables,  ability  to  "carry,"  and  ability  to  add  in  connection  with 
multiplication;  in  division  it  tests  knowledge  of  the  tables,  ability 
to  "carry"  in  short  division,  and  ability  to  solve  two  types  of 
examples  in  long  division,  the  one  involving  neither  "carrying" 
nor  "borrowing"  and  the  other  involving  both;  and  it  tests  the 
ability  to  apply  these  four  fundamental  operations  to  the  working 
of  examples  in  fractions. 

The  test  was  given  to,  and  results  secured  from,  834  classes  in 
the  schools  of  Cleveland  and  Grand  Rapids.  In  both  cities  the 
test  was  given  almost  entirely  by  the  teachers.  In  Cleveland  the 
teachers  were  inexperienced  in  giving  tests,  while  in  Grand  Rapids 
they  were  all  more  or  less  famiUar  with  the  Courtis  tests. 


CHAPTER  III 

GENERAL  RESULTS 

DETERMINATION  OF   STANDARDS 

In  order  that  the  test  may  be  of  the  greatest  educational  value 
it  is  necessary  that  standard  scores  be  determined  for  the  several 
grades  in  each  of  the  sets.  These  scores  must  of  course  be  deter- 
mined empirically,  that  is,  on  the  basis  of  what  children  actually  do. 

As  indicated  in  the  previous  chapter,  more  or  less  valid  results 
were  secured  from  two  large  school  systems,  Cleveland  and  Grand 
Rapids.  These  results  were  tabulated,  and  measures  of  central 
tendency  computed.  The  median  was  chosen  for  this  measure  for 
two  reasons:  (i)  since  it  is  not  disproportionately  affected  by 
an  extreme  case,  it  in  large  measure  eliminates  errors  due  to  over- 
timing or  undertiming;  (2)  the  median  is  easily  computed.  These 
two  facts  make  the  median  a  highly  desirable  average,  especially 
when  such  an  enormous  body  of  material  must  be  handled  as  is 
necessarily  the  case  in  the  survey  of  a  large  school  system. 

The  method  used  to  secure  a  final  average  score  for  each  of  the 
sets  of  the  tests  was  as  follows:  First,  the  median  of  each  of  the  90 
Cleveland  schools  (more  or  less  depending  on  the  grade)  in  a  par- 
ticular grade  was  found;  secondly,  the  median  of  these  medians  was 
computed  to  get  an  average  for  Cleveland  as  a  whole;  thirdly,  the 
same  thing  was  done  for  Grand  Rapids;  fourthly,  the  medians  of 
the  two  cities  were  averaged  to  get  tentative  standard  scores  for 
the  different  sets  of  the  test.  The  test  was  given  to  the  B  sections 
only  in  Cleveland  and  to  both  sections  in  Grand  Rapids,  but  since 
it  was  given  in  Cleveland  at  the  close  of  the  term  (June)  and  in 
Grand  Rapids  at  the  beginning  of  the  term  (February,  March),  in 
order  to  get  the  standard  score  the  results  from  the  lower  sections 
in  Cleveland  were  averaged  with  results  from  upper  sections  in 
Grand  Rapids. 

These  standard  scores  found  by  averaging  the  Cleveland  and 
Grand  Rapids  medians  appear  in  Table  II.    An  examination  of  the 


22 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


table  shows  that  the  average  scores  made  in  Set  A,  simple  addition, 
by  third-grade  pupils  in  the  time  allowance  (30  seconds)  was  13.4 
examples;  by  fourth-grade  pupils,  17. i  examples,  etc.  The 
absence  of  a  score,  as  in  the  earlier  grades  for  Sets  H,  K,  L,  N, 
and  O,  indicates  that  the  pupils  in  that  grade  were  unfamiliar  with 
the  type  of  operation  demanded. 


TABLE  n 

Averages  of  Median  Scores  in  Each  Arithmetic  Test  for  Grades  3-8. 
Cleveland  and  Grand  Rapids 


Set 

Grade 

3 

4 

s 

6 

7 

8 

A 

134 
8.9 
6.5 
6.3 
4-3 

2.0 
2.0 

17. 1 
12.8 

II. 7 

II. 4 

50 

45 
3-6 

21.9 
16.6 
14.8 
ISO 
S-9 

6.6 
S-i 
5-6 
1-7 
3-9 

5-6 
2.7 

3-4 
I.I 

24.9 

195 

16.8 

17.7 

6.7 

7-7 
55 
6.0 

31 
4-4 

7.0 

3-2 

41 
1.6 

Z-3 

27.0 
21. 1 
18.2 
20.3 

7-4 

9.1 
6.0 

7-7 
4.0 

51 

9-4 
3-8 

4-7 
1.9 

4-3 

28.9 
25.8 
19.9 
22.8 

c 

D 

E 

8.0 

F 

10.6 

G 

6.7 
8  6 

H 

I 

0.6 
1.9 

I.O 

30 

4.0 
1-7 
2.4 
0.8 

4-7 
6.1 

J 

K...t 

II. 4 
4-4 
S-4 
2.4 

S-2 

L 

M 

1-4 

N 

0 

It  is  freely  conceded  by  the  writer  that,  because  of  the  com- 
plexity of  the  test  and  the  difficulties  encountered  in  following  the 
time  allowances,  and  because  of  the  fact  that  the  test  was  quite 
largely  given  by  persons  with  little  or  no  training  in  testing,  it  is 
very  likely  that  many  errors  were  made  in  the  giving  of  the  test. 
Now  the  important  question  that  arises  is  the  nature  of  the  errors 
made.  If  they  were  of  a  compensating  sort — that  is,  if  it  were 
purely  a  matter  of  chance  whether  the  examiner  overtimed  or 
undertimed — ^the  errors  made  in  one  direction  were  offset  by  those 
made  in  the  other.  If,  on  the  other  hand,  the  errors  were  of  the 
cumulative  type — that  is,  if  for  any  reason  the  examiners  tended 
to  overtime  more  than  undertime,  or  vice  versa — the  errors  would 


GENERAL  RESULTS 


23 


not  offset  one  another,  and  an  error  would  enter  into  the  final 
results.  On  first  thought  it  would  seem  that,  since  the  tests  were 
being  given  in  connection  with  a  survey  to  determine  the  standing 
of  a  city  in  arithmetical  attainments,  as  well  as  the  relative  stand- 
ings of  the  individual  schools  within  the  city,  there  would  be  a 
tendency  for  the  teachers  to  overtime  rather  than  to  undertime. 

Some  light  may  be  thrown  on  our  problem  if  we  turn  to 
Table  III.  In  the  description  of  the  test  it  was  said  that  five  of 
the  sets — A,  B,  C,  D,  and  L — were  taken  over  from  Series  A  and  B 
of  the  Courtis  standard  tests.  It  is  therefore  possible  to  make  a 
comparison  between  the  Cleveland-Grand  Rapids  average  for  each 
of  these  sets  and  the  Courtis  standard  scores.  This  comparison 
is  made  in  Table  III  and  Diagram  i. 


TABLE  III 

Results  of  Cleveland  and  Grand  RAPros  Tests  Compared 
WITH  Courtis  Standards 


Score 

Set 

A 

B 

C 

D 

L 

3 

r Cleveland  and  Grand  Rapids. . 
\Courtis  Standard 

134 
130 

17. 1 
17.0 

21.9 

21 .0 

24.9 
25-0 

27.0 
29.0 

28.9 

31 -5 

8.9 
9-5 

12.8 
12.5 

16.6 
15-5 

19s 
19.0 

21. 1 
22.0 

25.8 
24-5 

6.5 
8.0 

II. 7 
115 

14.8 
150 

16.8 
18.5 

18.2 
20.5 

19.9 
22.5 

6.3 
8.0 

11. 4 

11. 5 

150 
150 

17.7 
18.5 

20.3 
22.0 

22.8 
24 -5 

4 

rCleveland  and  Grand  Rapids. . 
\Courtis  Standard 

1-7 
0.8 

5 

/Cleveland  and  Grand  Rapids. . 
\Courtis  Standard 

2.7 
2.0 

6 

r Cleveland  and  Grand  Rapids. . 
\Courtis  Standard 

3-2 
2.8 

7 

r Cleveland  and  Grand  Rapids. . 
\Courtis  Standard 

3-8 
?.  1 

8 

("Cleveland  and  Grand  Rapids. . 
\Courtis  Standard 

4-4 
4.0 

The  Courtis  standards  are  supposed  to  represent  June  attain- 
ments, while  the  Cleveland-Grand  Rapids  average  represents 
February  or  March  attainments,  almost  a  half-year  behind  the 
Courtis  standard.  A  word  should  be  said,  however,  about  Set  L. 
Very  little  reliance  can  be  placed  upon  this  comparison  because  the 


24 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


Courtis  standards  in  this  case  are  purely  tentative.    For  that 
reason  no  graphical  representation  is  made  of  the  comparison. 


Diagram  i. — Results  secured  from  Cleveland  and  Grand  Rapids  in  Sets  A,  B,  C, 
and  D  comptared  with  the  Courtis  standards. 


From  the  table  and  the  diagram  it  is  seen  that,  so  far  as  the 
seventh-  and  eighth-grade  attainments  in  the  four  sets  of  simple 
combinations  are  concerned,  the  Courtis  scores  are  higher  than  the 
Cleveland-Grand  Rapids  scores.    In  the  third  grade  the  same 


GENERAL  RESULTS  2$ 

difference  is  noted,  while  in  the  intermediate  grades  the  scores 
quite  closely  agree.  Two  facts  deserve  comment.  In  the  first 
place,  the  differences  on  the  whole  favor  the  Courtis  scores,  and  the 
differences  are  about  as  great  as  they  should  be  in  view  of  the  fact 
that  the  Courtis  scores  represent  an  advantage  of  almost  half  a 
grade.  The  presumption  is  strong,  therefore,  that  the  errors 
which  are  very  likely  to  have  accompanied  the  giving  of  the  test 
were  of  the  compensating  type  in  these  four  sets  at  least.  In  the  sec- 
ond place,  there  seems  to  be  a  characteristic  difference  in  the  forms 
of  the  two  curves  of  progress  from  grade  to  grade.  The  Courtis 
scores  indicate  uniform  progress  from  grade  to  grade,  while  the 
Cleveland-Grand  Rapids  scores  show,  though  not  emphatically, 
to  be  sure,  the  progress  to  be  less  rapid  with  each  successive  grade. 
In  other  words,  the  latter  tends  to  resemble  in  certain  respects  the 
t3^ical  learning  curve.  Thus  it  would  seem  that  the  question  has 
not  yet  been  answered  whether,  during  progress  through  the  grades, 
the  limits  which  are  ordinarily  set  to  improvement  through  practice 
are  completely  offset  by  the  maturing  of  the  pupil.  The  Courtis 
results  seem  to  indicate  that  these  Kmits  are  offset,  while  the  results 
from  Cleveland  and  Grand  Rapids  seem  to  point  to  the  contrary. 
To  return  to  the  question  of  error  in  the  final  result,  what  may 
be  said  concerning  the  reliability  of  the  scores  for  Sets  E  to  O 
inclusive?  Arguing  from  the  Courtis  standard  scores,  the  scores 
for  A,  B,  C,  and  D  seem  to  be  free  from  any  considerable  error. 
Through  experience  in  giving  the  test  the  writer  has  come  to  the 
conclusion  that  accurate  timing  is  more  difficult  in  connection  with 
these  first  sets  than  with  the  later  sets,  because  of  the  short  time 
allowances  in  the  former.  The  same  absolute  error  in  two  given 
cases  is  relatively  a  greater  error  where  the  time  allowance  is  small 
than  where  it  is  large.  It  would  at  least  seem  that  there  is  no  reason 
for  thinking  that  the  later  sets  were  not  given  as  accurately  as  the 
first  four.  To  this  last  statement  an  exception  should  possibly  be 
made  in  the  case  of  Set  H.  To  this  set  an  allowance  of  but  30 
seconds  is  given,  while  a  minute  is  given  to  each  of  the  two  preced- 
ing sets.  There  is  undoubtedly  a  tendency  for  the  examiner  to 
allow  a  minute  for  this  set  also,  so  that  there  is  probably  a  cause 
of  error  operating  in  Set  H  that  is  not  present  in  the  other  sets. 


26  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

From  the  foregoing  it  would  seem  that,  arguing  from  the 
Courtis  standards  to  Sets  A,  B,  C,  and  D,  and  from  these  four  sets 
to  the  remaining  sets,  with  the  possible  exception  of  H,  the  average 
scores  for  the  several  sets  made  by  the  pupils  in  the  lower  sections 
in  the  Cleveland  grades  and  the  upper  sections  in  the  Grand  Rapids 
grades  constitute  reliable  standards  for  midyear  attainments. 
These  standards  may  therefore  be  tentatively  accepted,  subject  of 
course  to  revision  as  returns  are  secured  from  other  cities. 

DERIVATION  OF  A  SYSTEM  OF  WEIGHTS 

It  is  desirable  for  certain  purposes  that  some  method  be  found 
of  equating  the  scores  made  in  the  different  sets  by  a  particular 
system,  school,  class,  or  individual  so  tiiat  a  single  score,  the  sirni- 
mation  of  the  scores  made  in  the  several  sets,  may  be  obtained  to 
indicate  general  attainment  in  the  "fundamentals."  In  order  to 
do  this,  a  unit  must  first  be  found  in  terms  of  which  the  score  made 
in  each  of  the  sets  may  be  stated. 

In  essence  the  equating  of  the  sets  resolves  itself  into  a  state- 
ment of  their  relative  difficulties.  There  are  two  factors  that  con- 
stitute the  criteria  of  difficulty:  the  first  is  speed,  the  second  is 
accuracy.  Since  accuracy  by  itself  means  almost  nothing,  since 
the  number  of  examples  attempted  likewise  means  but  little,  and 
since  the  number  of  examples  worked  correctly  in  a  given  time 
includes  a  measure  of  both  speed  and  accuracy,  it  seems  to  the 
writer  that  the  latter  is  as  valid  a  gauge  of  difficulty  as  any  that 
might  be  chosen.  Having  accepted  this  criterion,  the  equating  of 
the  sets  is  a  very  simple  matter.  A  second's  work  might  be  taken 
as  the  unit.  Then,  if  it  required  on  the  average  three  seconds  to 
work  one  example  and  two  seconds  to  work  another,  their  relative 
difficulties  would  be  as  three  is  to  two.  For  the  sake  of  conven- 
ience, however,  we  may  take  the  average  time  required  to  work  an 
example  in  one  of  the  sets  as  a  unit.  A  value  of  i .  o  is  then  given 
to  each  of  the  examples  worked  in  that  set,  with  values  for  the 
examples  of  the  other  sets  varying  inversely  as  the  speed  with 
which  they  can  be  worked. 

In  the  present  study  it  is  suggested  that  the  average  time 
required  to  work  an  example  in  Set  A  be  accepted  as  the  unit,  and 


GENERAL  RESULTS 


27 


that  each  example  correctly  worked  in  this  set  be  therefore  given  a 
value  of  1,0.  An  example  of  this  set  is  chosen  because  of  its  size 
and  stability.  It  is  a  smaller  quantity  than  any  other  unit  would 
be,  because  on  the  average  a  pupil  works  more  examples  of  this 
type  than  of  any  other.  It  is  more  stable  than  any  other,  there 
being  least  variation  from  individual  to  individual  in  the  records 
made  in  this  set.  A  second  suggestion  is  that  the  system  of  weights 
be  derived  from  the  records  made  by  eighth-grade  children.  It  is 
true  that  the  relative  difficulties  of  the  sets  are  not  the  same  from 
grade  to  grade.  Set  N  is,  for  example,  not  only  absolutely  much 
more  difficult  for  the  fourth  grade  than  for  the  eighth,  but  relatively 
much  more  difficult.  Of  course  it  is  possible  to  make  a  system  of 
weights  for  each  grade,  but  that  has  its  disadvantages.  The  thing 
desired  is  a  system  of  weights  that  will  show  progress  from  grade 
to  grade.  If  the  system  is  changed  with  every  grade,  there  is  no 
intelligible  relation  between  the  score  made  by  one  grade  and  that 
made  by  the  grade  above  or  the  grade  below.  Again,  it  would 
seem  that  the  relative  difficulties  of  two  sets  should  be  determined 
on  the  scores  made  by  individuals  who  have  attained  some  degree 
of  mastery  over  both  sets  rather  than  over  but  one.  Finally,  the 
system  of  weights  should  be  derived  from  the  eighth-grade  scores 
because  those  scores  represent  the  final  achievement,  under  the 
present  school  organization,  resulting  from  formal  training  in 
arithmetic. 

In  Table  IV  the  system  of  weights  is  shown  and  the  method  of 
deriving  them  is  indicated.    In  the  first  horizontal  column  of  the 

TABLE  IV 

Derivation  of  System  of  Weights 


Set 


A 

B 

c 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

28.9 

2S.8 

19.9 

22.8 

8.0 

10.6 

6.7 

8.6 

4-7 

6.1 

ir.4 

4-4 

S-4   2-4 

30 

30 

30 

30 

J° 

60 

60 

30 

60 

120 

120 

180 

180   180 

28.9 

2S.8 

19.9 

22.8 

8.0 

S-3 

335 

8.6 

2. 35 

I  S3 

2.8s 

.73 

.90   40 

I.O 

,1.12 

1-45 

1.27 

3.61 

5  45 

8.63 

3.36 

12.3 

18.9 

10. 1 

39S 

32.1 

72.2 

Eighth-grade  score 

Time  allowance  in  seconds 

Score  per  30  seconds 

Weight  (relative  difficulty) 


S.a 
180 
.87 
33-3 


table  are  the  average  eighth-grade  scores  for  the  15  sets;  in  the 
second  are  the  time  allowances  in  seconds;  in  the  third  is  the  aver- 
age score  per  30  seconds  for  each  set;  and  in  the  fourth  are  the 


28 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


weights,  or  measures  of  relative  difficulty.  However,  since  the 
time  allowance  varies  from  set  to  set,  the  system  of  weights  as  pre- 
sented in  this  table  requires  some  revision.  For,  as  the  weights  now 
stand,  each  of  the  sets  with  time  allowances  of  3  minutes  has  just 
six  times  as  much  influence  in  determining  the  total  score  as  has 
any  one  of  the  sets  with  time  allowance  of  30  seconds.  It  is  there- 
fore necessary,  in  order  that  each  set  may  have  the  same  influence 
on  the  total  score  as  any  other  set,  to  modify  the  weights  to  that 
end.    This  revised  system  of  weights  appears  in  Table  V.    The 

TABLE  V 
Equation  of  Time  Allowances 


Set 

A 

B 

C 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

Weight  (relative  difficulty) 
Time  allowance  in  seconds 
Weight  after  equating 
time  allowances 

I.O 

1.0 

X.I3 
30 

1. 13 

I-4S 
30 

1-45 

1.27 
30 

1.27 

3.61 
30 

3.61 

60 

a. 73 

8.63 
60 

431 

3.36 
30 

3.36 

13.3 
60 

6. IS 

l8.9 
120 

4-73 

10. 1 
1 30 

2. S3 

39S 
180 

6.s8 

33  I 
180 

5-3S 

73.3 
180 

12.0 

33-3 
180 

S.S6 

same  result  would  have  been  secured  if  in  Table  IV  the  time  allow- 
ance had  been  neglected  entirely  and  the  weights  computed  on  the 
average  scores  as  they  stood.  Such  a  method,  however,  would  have 
been  misleading.  As  shown  by  the  two  steps  taken  in  evolving  the 
systems  of  weights,  it  now  does  two  things:  (i)  it  equates  the 
examples  on  the  basis  of  difficulty,  and  (2)  it  equates  the  time 
allowances  of  the  several  sets. 


THE  USE   OF   THE  TEST 

In  a  previous  chapter  it  was  pointed  out  that  the  test  was 
evolved  for  the  purpose  of  diagnosing  weaknesses  of  one  sort  and 
another  in  school  systems,  schools,  classes,  and  individuals.  The 
method  by  which  the  test  may  be  used  for  doing  this  thing  will  here 
be  demonstrated. 

We  shall  first  make  some  comparisons  between  two  large  city 
systems — Cleveland  and  Grand  Rapids.  By  using  the  system  of 
weights  just  described,  it  is  possible  to  get  a  single  score  to  represent 
the  arithmetical  attainments  of  each  grade  for  each  of  these  two 
cities.  These  scores  are  given  in  Table  VI  and  graphically  repre- 
sented in  Diagram  2. 


GENERAL  RESULTS 


29 


Turning  to  the  diagram,  we  see  that  there  are  considerable 
dififerences  between  the  two  curves,  especially  in  the  lower  grades. 


TABLE  VI 

Comparison  op  Total  Scores  Made  by  Grades  3-8  m  Cleveland  and 
Grand  Rapids 


Grade 

3 

4 

5 

6 

7 

8 

Cleveland 

90 
22 

181 
132 

260 
243 

318 
32s 

370 
383 

4^0 

Grand  Rapids 

439 

S  no 

3  **o 


The  superior  attainment  of  the  Cleveland  children  in  the  lower 
grades  indicates  relatively  greater  stress  on  arithmetic  in  this 
period.  Work  in  arithmetic  is  begun 
earlier  in  Cleveland  than  in  Grand 
Rapids,  but  this  initial  advantage  is 
not  maintained.  From  the  showing 
that  the  latter  city  has  made,  the 
conclusion  would  seem  to  be  justi- 
fied that  a  large  expenditure  of  time 
on  arithmetic  in  the  lower  grades  is 
of  comparatively  little  importance 
in  securing  high  attainment  in  the 
eighth  grade. 

However,  this  is  a  very  general 
result  and  by  itself  is  of  little  value 
because  its  meaning  is  vague  and 
uncertain.  The  important  thing  for 
either  city  to  know  is  its  weak  points.  The  detailed  records  by 
which  this  is  possible  are  found  in  Tables  VII  and  VIII 
and  Diagrams  3,  4,  5.  In  Diagram  3  there  appears  a  graphic  com- 
parison of  the  records  made  by  the  pupils  of  the  two  systems  in 
each  of  the  four  sets  in  addition.  A,  E,  J,  M.  The  first  glance  at 
these  curves  shows  that  the  relations  between  the  two  cities  are 
about  the  same  here  as  indicated  by  the  general  results,  viz., 
superiority  of  Cleveland  in  the  lower,  and  superiority  of  Grand 


Diagram  2. — Comparison  of 
total  scores  made  by  Grades  3-8 
in  Cleveland  and  Grand  Rapids. 


30 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


Rapids  in  the  upper,  grades.    In  the  simple  addition  combinations, 
Set  A,  there  is  little  difference  between  the  two  systems,  except  in 

TABLE  VII 

Median  Scores  in  Each  Arithmetic  Test  foe  "B"  Sections  or 
Grades  "3-8,  Cleveland 


Grade 

3 

4 

s 

6 

7 

8 

A 

B 

134 
9  3 
6.S 
6.3 
4-3 

2.0 
2.0 

17.8 

134 
12.0 
12.4 

5-3 

4-9 
3-9 

22.2 
17.2 
ISS 

15-7 

6.3 
6.7 

5-2 

50 

2.0 

4.0 

6.8 
2.5 

3-2 

1-3 

24.8 
19.8 
16.6 
18.5 
6.8 

7.5 
5-5 
5-5 
3-1 
4-4 

8.5 
2.8 
3.8 
1-7 
3-1 

26.7 

21.5 
17.7 
20.8 

7-5 

8.6 
5-9 
7-7 
4.0 

4  9 

10. 1 
3-2 
4.4 
2.0 
41 

275 
26.0 

C 

19.0 

22.5 

7.8 

D 

E 

F 

10. 1 

G 

6.6 

H 

8.5 
4.7 
5-7 

12. 5 
3-9 
5-1 
2.6 

I 

0.6 
1.9 

I.I 

3-2 

4.0 
1-7 

2.5 

0.8 

J 

K 

L 

M 

1-4 

N 

0 

5-5 



♦  TABLE  Vni 

Median  Scores  in  Each  Arithmetic  Test  for  Grades  3-1-8-2.  Grand  Rapids 


Test 

Grade 

3-1 

3-a 

4-1 

4-a 

S-i 

S-2 

6-1 

6-a 

7-1 

7-2 

8-1 

8-a 

A 

B 

c 

II. 8 
6.3 

13-4 
8.4 

13.6 
9.1 
71 
6.9 
4-1 

2.8 
2.2 

16.4 

12. 1 

"3 

10.4 

4.6 

41 
3-3 

20.3 
14-7 
13.7 
12.5 

5-2 

6.0 
4.0 

21.5 

15-9 
14.0 

14.3 
5-4 

6.5 
4.9 
6.3 
1.4 
3-7 

4-3 
2.9 
3-6 
0.8 

22.8 
16.8 
15s 
15. 5 
6.0 

7.1 
5.3 
6.2 

2.3 
4.1 

5-4 
3-3 
4-3 
I.I 

3-5 

25.0 
19. 1 
17.0 
16.9 
6.6 

8.0 
5-6 
6.5 
3.0 
4-5 

6.5 
3.6 
4-5 
1-4 

3.6 

26.5 
21.3 
17.7 
18.4 
7.2 

9.3 
6.1 
9.0 
3-8 
5-4 

7-5 
4-3 
4-9 
1.7 
3-9 

273 
20.7 
18.8 
19.7 
7.2 

9.6 
6.1 
7.8 
4-1 
5-3 

8.8 

4.5 
5.0 
1.8 
4.6 

295 
22.8 

49-3 

20.5 

7.8 

10.3 
6.7 
8.6 
4.0 
5-7 

9-7 
4  9 

5-7 
2.0 

55 

30.3 
25.5 
20.7 
23.0 
8  I 

D 

E 

F 

G 

6  8 

H 

8  8 

I 

0.7 

0.9 
2.8 

1-3 
3-4 

3.0 

2-3 

3.0 
0.7 

4-7 
6.5 

10.3 
4  9 
5-7 
23 
4.8 

J 

K 

L 

M 

2-3 

N 

0 

GENERAL  RESULTS 


31 


the  eighth  grade,  where  Grand  Rapids  clearly  takes  the  lead.  In  the 
short-column  addition  Grand  Rapids  is  comparatively  weak,  while 
in  the  more  complex  types  of  addition.  Sets  J  and  M,  involving  the 


Diagram  3. — A  comparison  of  records  made  in  Sets  A,  E,  J,  and  M  (addition)  by 
Grades  3-8  in  Cleveland  and  Grand  Rapids. 

bridging  of  the  attention  spans  and  "carrying,"  the  superiority  of 
that  city  is  beyond  question. 

In  Diagram  4  we  have  the  same  comparisons  for  the  three  sets 
in  multiphcation,  C,  G,  and  L.  The  facts  here  tend  to  confirm  the 
previous  statements  concerning  the  relative  standings  of  the  two 


32 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


cities.  It  should  be  added,  however,  that  the  characteristic  differ- 
ence is  somewhat  accentuated  in  Set  L,  the  multiplication  of  four- 
place  numbers  by  two-place  numbers,  a  type  of  operation  in  which 
Cleveland  appears  to  be  particularly  weak. 

Passing  to  Diagram  5,  conditions  are  found  to  be  completely 
reversed.  The  test  has  revealed  the  weakness  of  the  children  of 
Grand  Rapids  in  the  fundamentals — division.  In  every  one  of  the 
sets  in  division,  D,  I,  K,  and  N,  the  Cleveland  scores  appear  to 
advantage,  but  it  is  in  the  two  sets  in  long  division,  K  and  N,  that 
this  relative  excellence  is  most  marked.  The  Grand  Rapids  chil- 
dren have  not  mastered  the  technique  of  long  division. 


DuGRAM  4. — Comparison  of  records  made  in  Sets  C,  G,  and  L  (multiplication) 
by  Gjrades  3-8  in  Cleveland  and  Grand  Rapids. 

In  subtraction  and  fractions  the  characteristic  difference  is  again 
apparent;  comment  on  the  records  made  in  the  sets  involving  these 
operations  is  therefore  unnecessary.  Enough  has  been  said  to  indi- 
cate how  the  weaknesses  in  a  school  system  may  be  detected. 

As  an  instrument  in  the  hands  of  the  supervisor  the  test  is  of 
decided  value.  For  the  purpose  of  diagnosing  class  weaknesses  the 
records  made  in  the  several  sets  may  be  graphically  represented  as 
in  Diagram  6,  a  form  of  graph  used  by  Mr.  Courtis.  In  this  dia- 
gram the  horizontal  lines  represent  the  15  sets  of  the  test,  the  ver- 
tical lines  the  grades,  and  the  irregular  lines  the  records  made  by 
the  three  classes  in  Grade  6-2  in  three  Grand  Rapids  schools. 
Turner,  Lafayette,  and  East  Leonard.  The  figures  at  the  points 
of  the  intersection  of  the  lines  are  the  average  scores  for  Cleveland 
and  Grand  Rapids  made  in  the  indicated  sets  by  the  indicated 
grades.    Thus,  if  a  particular  sixth-grade  class  should  make  scores 


GENERAL  RESULTS 


33 


that  would  exactly  agree  in  every  case  with  this  general  average,  the 
graphical  representation  of  the  record  made  by  that  class  would 
coincide  with  the  heavy  black  vertical  line  representing  the  sixth 
grade  in  the  diagram.    Deviations  from  this  line  indicate  deviations 


Craeft-S 


4         5         6 
£ra(/es 


Diagram  5. — A  comparison  of  records  made  in  Sets  D,  I,  K,  and  N  (division)  by 
Grades  3-8  in  Cleveland  and  Grand  Rapids. 


either  above  or  below  the  Cleveland-Grand  Rapids  average,  depend- 
ing on  whether  such  deviations  are  to  the  right  or  to  the  left. 

Now  on  examining  the  diagram  there  is  little  tendency  observed 
on  the  part  of  these  three  sixth-grade  classes  represented  to  follow 
the  general  average  as  indicated  by  the  heavy  vertical  line.    The 


34 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


East  Leonard  School,  represented  by  the  broken  line,  is  seen  to  be 
doing  work  of  a  relatively  high  order,  since  it  drops  below  the 
average  in  no  set.  In  Sets  C  (simple  multiplication  combinations), 
H  (fractions),  and  0  (fractions)  this  class  is  especially  strong,  while 


Tu  rae  f       

Z  *  /«/«  '^e  *  *  •  •  — * 

C  LAOnarJ   —  —  —  — 


DiAGSAM  6. — A  comparison  of  the  records  made  in  each  of  the  fifteen  sets  by  the 
sixth  grades  in  three  Grand  Rapids  schools — Turner,  Lafayette,  and  East  Leonard. 

its  greatest  weaknesses  seem  to  be  in  division,  I,  K,  and  N,  the 
typical  Grand  Rapids  weakness.  The  Turner  School,  represented 
by  the  solid  line,  is  seen  to  be  of  an  entirely  different  type.  Its  work 
is  consistently  of  a  low  order,  being  below  the  average  in  every  set. 
The  other  class  gives  evidence  of  very  poor  supervision.    Note  the 


GENERAL  RESULTS 


35 


erratic  character  of  the  curve.  The  class  is  very  weak  in  short- 
column  addition,  E,  and  exceptionally  strong  in  the  addition  of 
5  four-place  numbers;  very  poor  in  one  type  of  fractions,  H,  and 
very  good  in  another,  0;   but  good  in  both  of  the  more  complex 


nS. 


Diagram  7. — A  comparison  of  the  records  made  in  each  of  the  fifteen  sets  by 
two  sixth-grade  pupils  in  the  same  class  in  Grand  Rapids. 

types  of  multiplication,  G  and  L.  Many  more  facts  could  be 
pointed  out,  but  these  will  suffice  to  show  how  the  test  may  be  used 
to  enable  the  supervisor  to  discover  class  weaknesses. 

We  come  now  to  the  use  of  the  test  by  the  teachers  in  studying 
the  peculiarities  of  the  individual  pupil.    In  Diagram  7,  identical 


36  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

in  form  with  the  immediately  preceding  diagram,  is  presented  the 
records  of  two  sixth-grade  pupils  in  the  same  class.  The  pupil 
R.  S.  is  seen  to  be  weak  in  everything  except  Sets  C  and  L,  two 
sets  in  multiplication.  The  pupil  E.  A.,  on  the  other  hand,  is 
strong  in  everything  except  the  multiplication  tables,  C,  short- 
column  addition,  E,  short  division,  I,  the  complex  addition,  M,  and 
fractions,  O.  With  this  record  before  her  the  teacher  would  be 
able  to  direct  the  pupils'  time  and  energy  into  the  needed  channels 
and  expend  her  own  to  a  rational  end. 

DISTRIBUTIONS 

A  measure  of  central  tendency,  when  used  to  represent  a  group, 
is  always  subject  to  the  criticism  that  it  is  a  single  measure  and 
gives  no  idea  of  the  variations  from  this  central  tendency  of  the 
individuals  composing  the  group.  It  is  therefore  necessary,  in 
order  that  we  may  get  a  complete  picture  of  the  scores  made  by 
all  the  individuals  composing  the  group,  to  present  the  entire  dis- 
tribution of  the  individual  records. 

If  the  distribution  is  to  be  valid,  it  is  absolutely  essential  that 
thepe  be  no  mistakes  made  in  the  timing  of  the  individual  pupils 
forming  the  distribution.  When  determining  a  measure  of  central 
tendency,  as  pointed  out  earlier  in  the  chapter,  it  is  merely  necessary 
that  no  cumulative  error  be  made,  since  the  chance  errors,  occurring 
as  often  on  the  one  side  of  the  central  tendency  as  on  the  other, 
offset  one  another.  In  the  case  of  the  distribution,  on  the  other 
hand,  it  is  obvious  that  these  chance  errors  flatten  out  the  distri- 
bution, since  through  error  individual  scores  would  be  shifted 
to  the  one  side  or  the  other  of  this  central  tendency  and  thus 
decrease  the  actual  frequency  at  that  point.  For  this  study  of 
distribution,  therefore,  only  the  records  made  by  pupils  examined 
by  the  writer  will  be  used.  The  number  of  pupils  in  each  grade 
thus  tested  in  the  schools  of  Grand  Rapids  appears  in  Table  IX. 
The  number  of  cases  is  not  great  for  any  grade,  but  the  records  are 
accurate. 

In  Tables  X,  XI,  XII,  and  XIII  are  presented  the  distributions 
for  each  grade  in  each  of  the  four  sets  in  addition,  A,  E,  J,  and  M. 


GENERAL  RESULTS 


37 


It  will  be  noted  that  the  distributions  have  been  reduced  to  a  per- 
centage basis  in  order  that  the  results  from  grade  to  grade  may 
be  strictly  comparable. 

TABLE  DC 

Number  of  Pxjpils  in  Each  Grade  Tested  by  the  Writer  in  the 
Grand  Rapids  Schools 


Grade 

3-1 

3-2 

4-1 

4-2 

s-i 

5-2 

6-1 

6-2 

7-1 

7-2 

8-1 

8-2 

Total 

Pupils 

31 

30 

38 

123 

70 

108 

166 

193 

13s 

174 

176 

132 

1.376 

Since  the  interpretation  of  the  facts  as  they  appear  in  the  tables 
is  comparatively  difficult,  the  same  facts  for  the  upper  sections  of 
the  grades  are  presented  graphically  in  Diagram  8.  The  diagram 
is  so  constructed  that  movement  down  the  graph  from  top  to 
bottom  is  in  line  with  progress  through  the  grades,  while  movement 
from  section  to  section  across  the  diagram  means  movement  from 
a  simple  type  of  operation  to  a  more  complex  type.  If  this  expla- 
nation be  kept  in  mind  some  very  interesting  and  significant  rela- 
tions may  be  noted  in  addition  to  the  general  fact  that  the  curves 
on  the  whole  indicate  a  normal  distribution,  that  is,  the  largest 
number  of  cases  being  near  the  central  point  of  the  distribution 
with  a  symmetrical  decrease  on  either  side  to  zero. 

A  fact  very  clearly  brought  out  by  the  diagram  is  that  in  each 
of  the  four  sets  the  distribution  curve  tends  to  become  flattened 
with  progress  through  the  grades.  With  few  exceptions  the  curves 
for  both  the  "attempts"  and  the  "rights"  become  flattened  with 
each  successive  grade.  This  means  that  the  training  received  in 
the  schools  tends  to  accentuate  individual  differences  rather  than 
the  contrary.  While  there  is  general  progress  of  a  particular  type 
through  the  grades,  the  individuals  of  the  grade  in  many  instances 
depart  from  this  typical  rate  of  progress.  The  methods  used  work 
well  with  some  individuals,  but  scarcely  at  all  with  others;  and  it 
is  probably  true  that  with  the  same  training  the  bright  pupil,  while 
making  relatively  the  same  progress,  makes  absolutely  much  more 
than  the  dull  pupil.  Thus  the  individual  variation  is  increased, 
while  progress  is  made  in  both  cases. 


38 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


Another  fact  brought  out  in  the  diagram  is  that  the  curves  also 
become  flattened  as  we  proceed  from  left  to  right  across  the  graph, 
that  is,  from  the  less  complex  to  the  more  complex  types  of  addition. 


Si| 


*' 


S»^  A 


Att- 


Strt  J 


Cf^J*  M-t 


SnJe  t-i 


it 


6.  is 


£i></e  y-t 


CriL/l  t-i 


£ra^9  ^^ 


if 


DiAGSAM  8. — A  comparison  of  the  distribution  of  "attempts"  and  "rights"  in 
four  sets  in  addition  (A,  E,  J,  M)  for  Grades  3-8. 


This  is  not  especially  true  in  going  from  Set  J  to  Set  M,  but 
it  must  be  remembered  that  these  are  two  different  t5^es  of  addition, 
the  one  involving  the  bridging  of  the  attention  spans  and  the  other 
"carrying."    Which  of  these  two  types  is  the  more  complex  it 


GENERAL  RESULTS 


39 


B 

< 

M 

X 

2 

H 

m 

hJ 

f^ 

m 
< 

s 

H 

(Jk 

01    W    1-1    M    ci    w 


►-iM<*^cO«MC««>l« 


rj.  Tf  «r»  to  N.*0  <0  t^  N   w  U-,  Tl- 


^   : ^   :^   : ^   :^   :i2   :is   :4S   :£J   '^   :^   ■  ^   '■ 

ii;43  a)rfl  (u  -^  a  JS  (u  •O  qjrfl  oj-Q  iujC  a>-4  iu>C  u^C  (UJ3 
-<->   bO-'-'   bC-»->   bC-t-i   &£-»->   bC-'-'   bC-t-'   &£-<-'   bC-M   bC-i-i   bp -^   bC-^   bO 


to      vO 


W  M  M 

r-      /»      o6 


00 


40  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


H 
H 
CO 


O 


B 

<: 

W 

I— 1 

X 

j5 

H 

CO 

h4 

ri 

(14 

0»     M    «»^  «*5      •     M 


"O  <*5  r*  «  N  « 


►<     •  ci|  c<  Ki  w  CT)  «  ^h\o  >0  i^i  «^  <0 


vp»  It  e>(   t1   «»i  t<3  «^  «*5^  vo  t^  to 


00  «n>o    •  to 


^M»0    'fO    -WClWKiMt-ii-t 


11  8:2  g|  8:1  e|  S;!  B^  n^  aj  a^^ 

4>^  oirC  Vp4  <u43  u,JCi  a><4   (UrQ  (U>ci  <u^   m^ 

<:  rt  <  S  <  S  •<  S  <:  S  <  S  <  S  <  S  <  S  <  S 


W  C<  M  M  W 

^        •^        10        ilr>      ><b 


M  ft 

I  I  I  I  I 

>0         t^       t^       00       00 


GENERAL  RESULTS 


41 


would  be   diflScult   to   say.    However,  in   the   other   cases   the 
statement  is  certainly  true.     This  would  indicate  less  individual 

TABLE  XII 

DlSTEIBirTION  OF   lOO  PUPILS  IN  EACH  GrADE — SeT  J 


Score 

0 

I 

3 

3 

4 

5 

6 

7 

8 

9 

10 

XI 

13 

13 

14 

/Attempts. 
^-^Rights  ... 

21 

2 

18 

18 
21 
10 
17 

4 
18 

2 
10 

4 
13 

I 
12 

30 

23 
20 
27 

17 
20 
10 
28 
JO 
19 
P 
13 

3 
8 
2 

14 
5 

16 

JO 

8 
'/J 
14 
31 
21 
22 
21 

^r 
19 

22 
ij 

24 
10 

IS 

JO 

18 

14 
8 

■r7 
13 
i5 

9 

28 

II 

2-^ 
18 
24 

IS 
^7 
21 

Jp 

19 

17 

8 

5 

7 
2 

8 
2d 

8 
18 

7 
16 

8 
23 
13 
JP 
10 
i5 
16 

I 
2 

I 

9 

I 

IT 

6 

15 

8 

^4 

9 

2J 

14 
18 

IS 

22 

II 

f  Attemots . 

3 

5"^1  Rights.... 

9 

"s 

17 

I 

12 

/Attempts. 
5  ^\Rights.... 

3 

3 
7 
S 
5 
3 
5 
3 
8 
6 

J4 
8 

II 
3 

I 

I 

^-A^^t:. 

3 
2 

3 
2 

5 
2 

7 
3 

8 

4 
P 
3 

2 

7 

^  „/ Attempts. 

^^\Rights.... 

1  Attemots . 

3 

1 
2 
2 

3 

I 

5 

I 

I 

3 

7 

3 

I 

2 
I 

I 
2 

X 

'"^Rights.... 

4 

8 

fAttemots . 

I 

z 

7'^\Rights.... 
0    fAttemots . 

I 

I 

6 

I 
7 

12 

z 

^-^Rights     . 

2 

2 

4 

I 

10 

0  „/ Attempts. 
^nRights.... 

2 

TABLE  XIII 
Distribution  of  100  Pupils  in  Each  Grade — Set  M 


Score 


/Attempts. 
4"21  Rights. 


/Attempts . 
"M  Rights.... 


/Attempts . 
5-^1  Rights..., 
,  /Attempts. 
^^1  Rights.... 
^  /Attempts. 
HRights.... 

/Attempts. 
7-^1  Rights.... 

J  Attempts . 
7-21  Rights.... 
Q  ^/Attempts. 
^■M  Rights.... 
Q  „ /Attempts. 
^-^Rights  ... 


16 


5 
22 

4 
14 


IS 


19 
21 

17 

33 

6 

16 

II 

2 

14 


36 
20 
Id 

14 
22 
26 
10 
24 
10 
21 
8 
II 

19 

3 

12 

19 


33 
14 
27 
19 
22 
21 
16 
21 
^5 
19 
^5 
18 

9 
IS 

7 
16 


IS 
7 

25 
7 

22 

9 

30 
16 

30 
19 
25 
IS 
22 
16 
i5 
20 
24 
18 


2 

2 

z6 

7 
21 
II 

23 
10 
18 
IS 
25 
18 

29 
21 

23 

18 


4 

2 

<J 

2 

II 

6 

z6 

II 

Id 

10 

Jp 

7 

20 


2 
d 

zi 
4 
7 
2 

12 


2 

I 
I 

I 

3 

2 
d 

2 

3 

10 

2 


42 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


variation  on  the  simple  addition  examples  than  on  the  more 
complex. 

The  character  of  the  relation  between  the  curves  for  the  "rights" 
and  for  the  "attempts"  is  a  third  matter  deserving  attention.  On 
the  average  the  curve  for  the  "rights"  is  flatter  than  that  for  the 
"attempts."  This  is  emphatically  true  in  Sets  J  and  M,  the  more 
complex  types.  Thus  there  is  less  tendency  among  the  pupils  to 
vary  in  the  number  of  examples  attempted  than  in  the  number 
solved  correctly.  This  is  probably  explained  by  the  fact  that  the 
number  of  examples  attempted  is  controlled  quite  largely  by  the 
physical  limitations  on  speed,  since  the  character  of  the  operation 
in  each  of  these  types  of  examples  is  familiar  to  all  the  pupils.  In 
working  the  examples  correctly,  on  the  other  hand,  another  factor 
is  involved,  and  that  is  the  factor  of  right  and  wrong  associations- 
A  more  strictly  mental  limitation  is  here  added  to  the  physical 
limitation  just  mentioned. 

TABLE  XIV 

Distribution  op  ioo  Pupils  in  Each  Grade — Set  L 


Score 


.      /Attempts. 


'1  Rights. 

/Attempts. 
5"21  Rights... 
>.  /Attempts. 
^'JRights  .. 
e,  /Attempts. 
^-^Rights  .. 

/Attempts. 
7"^1  Rights... 

/Attempts. 
7*^1  Rights. . . 
Q  , /Attempts. 
^"M  Rights... 
Q  ^/Attempts. 
8-^Rights  .. 


26 
12 

4 
3 
4 


8 

20 


i6 

2 

13 

I 

12 

I 

lO 


27 
26 

14 

28 

8 
19 

3 
17 

2 

17 

2 

lO 

2 

7 

I 

II 


37 

21 
32 
25 
22 
26 

14 

24 

II 

20 

9 
i8 

3 

17 

6 

20 


26 

6 

38 
12 

31 
19 

34 
22 
29 

25 

17 
22 

13 
24 
15 
23 


2 

I 
II 

4 
20 

12 

2Q 

14 
21 
10 
22 

19 
21 
21 
23 
14 


2 

2 
II 

4 

12 

7 
19 
II 

23 
IS 
27 
16 

24 
17 


3 
17 

8 
22 

7 
15 

6 


10 
I 

II 
2 

16 
4 


Since  the  foregoing  characteristics  are  not  peculiar  to  the  dis- 
tributions in  addition,  but  are  common  to  the  distributions  in  each 
of  the  other  three  fundamentals  as  well,  it  is  not  necessary  to  present 
tables  and  graphs  setting  forth  the  distributions  in  these  three 


GENERAL  RESULTS 


43 


operations.  We  have  an  entirely  different  proposition  in  the  case 
of  fractions.  For  this  reason,  therefore,  Tables  XIV  and  XV  are 
accompanied  by  Diagram  9,  which  graphically  portrays  the  facts 
found  in  the  tables.  In  the  first  table  appear  the  distributions  for 
the  several  grades  in  Set  L,  multiplication;  in  the  second  the  dis- 
tributions in  Set  O,  fractions. 


TABLE  XV 

Distribution  of  icx3  Ptn>iLS  in  Each  Grade — Set  O 


Score 


/Attempts. 
S'^l  Rights.... 
,  J  Attempts . 
^"M  Rights.... 
£  J  Attempts . 
^Rights  ... 

!  Attempts . 
Rights.... 
Attempts . 
''^1  Rights.... 
Q  J  Attempts . 
^■'1  Rights.... 
Q  ./Attempts. 
^"^\Rights.... 


13 


7 
31 

3 
16 

3 
22 

2 

13 
I 

10 
I 

10 
2 
9 


20 
28 

8 

18 

6 
18 

7 
23 

3 
22 

3 
19 

4 
19 


II 
I 
16 
13 
15 
8 

7 
5 
6 
8 

7 
10 

7 
II 


II 

4 

II 

I 

9 

9 

II 

5 
9 
8 

9 

S 


15 


27 

'18 

I 

14 

I 


The  diagram  is  of  the  same  order  as  the  previous  one  and  there- 
fore requires  no  explanation.  The  similarity  between  the  curves 
for  Set  N  and  those  for  the  sets  in  addition  just  discussed  is  appar- 
ent. Let  us  therefore  turn  at  once  to  a  comparison  of  the  curves 
of  this  set  and  of  Set  O.  Perhaps  the  most  obvious  feature  in  the 
comparison  is  the  relation  between  the  curves  for  the  "rights"  and 
the  curves  for  the  "attempts"  in  Set  O.  Here,  in  direct  contrast  to 
the  sets  in  the  fundamentals,  the  curves  for  the  "attempts"  present 
a  much  more  flattened  appearance  than  the  curves  for  the  "rights." 
This  would  seem  to  indicate  that,  whereas  in  the  fundamentals  the 
knowledge  of  the  character  of  the  operation  to  be  performed  was 
common  property  for  practically  all  the  pupils,  in  fractions  the 
character  of  the  operation  is  not  known  by  all.  To  those  familiar 
with  the  method  of  handling  fractions,  or  to  those  who  think 
themselves  familiar  with  it,  it  is  a  simple  matter  to  attempt  a 


44 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


Ms: 


Se^L 


S&€iXI 


Crad£  S-Z 


droLoe.  6-Z 


-^ 


CraJe.  7-?. 


>2AJ 


GrOiJ^  8-Z 


.35 


/\/o.o/£'^c(rn/fles  No,  <?/  dcamptes 

DxAGRAii  9. — A  comparison  of  the  distribution  of  "attempts"  and  "rights"  in 
Set  L  (multiplication)  and  Set  O  (fractions)  for  Grades  5-8. 


GENERAL  RESULTS 


45 


large  number  of  the  examples.  This  is  not  true  of  the  fundamentals. 
For  instance,  take  an  example  in  long  division.  Even  though  the 
method  of  working  such  an  example  is  perfectly  familiar,  it  requires 
considerable  time  to  work  it  because  it  is  a  long  process.  Speed 
can  be  developed  only  through  much  practice  by  making  a  large 
number  of  reactions  quite  automatic.  The  actual  process  involved 
in  working  an  example  in  fractions,  such  as  is  found  in  Set  O,  is,  on 
the  other  hand,  a  relatively  short  one.  Now,  since  so  far  as  attempt- 
ing the  examples  is  concerned  it  is  just  about  as  easy  to  attempt 
one  of  the  examples  as  another,  those  pupils  who  are  familiar  with 
the  method  of  solving  fractions  attempt  a  large  number,  or  all  of 
them.  Those,  on  the  other  hand,  who  are  unfamiliar  with  the 
method  are  able  to  attempt  but  a  few.  In  this  way  the  curve  for 
"attempts"  becomes  flattened.  The  curve  for  "rights"  is  less 
flattened  because  of  the  composition  of  the  test  set.  As  will  be 
pointed  out  later,  the  examples  in  the  multiplication  and  division 
of  fractions  are  easier  than  the  other  two  types.  This  causes  the 
distribution  of  "rights"  to  be  largely  confined  to  six  examples. 
Since  there  is  no  such  factor  operating  to  narrow  down  the  distri- 
bution of  "attempts,"  the  curve  for  the  "rights"  is  elevated  in 
comparison. 

Another  fact  indicated  by  the  diagram  which  bears  somewhat 
on  this  same  matter  is  the  increase  of  the  percentage  of  pupils 
attempting  all  the  examples  in  Set  O  up  to  Grade  7-2  and  then  a 
decrease  in  the  percentage  to  Grade  8-2.  An  examination  of 
Table  XV  gives  further  evidence  on  this  same  point.  It  is  seen 
that  there  is  a  constant  increase  in  this  percentage  from  Grade  5-2 
through  Grades  6-1,  6-2,  and  7-1  to  Grade  7-2,  where  the  maximum 
of  27  per  cent  is  reached.  Then  there  is  a  decrease  to  18  per  cent 
in  Grade  8-1,  and  a  further  decrease  to  14  per  cent  in  Grade  8-2. 
This  is  a  very  significant  fact.  An  inspection  of  the  work  actually 
done  by  the  pupils  on  this  set  indicates  a  tendency  among  them  to 
substitute  various  "easy"  methods  for  the  correct  methods  in  work- 
ing the  examples.  For  example,  a  pupil  may  add  two  fractions  by 
adding  their  numerators  and  their  denominators.  This  takes  less 
time  than  the  right  method.  Thus,  by  substituting  invalid  for 
valid  methods  the  pupil  is  enabled  to  complete  the  set  in  a  relatively 


46 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


short  time.  As  the  pupil  matures  he  gradually  develops  greater 
speed,  and  this  probably  accounts  for  the  increase  in  the  number  of 
pupils  attempting  all  the  examples  of  the  set  up  to  Grade  7-2.  The 
decrease  from  this  point  on  is  probably  due  to  the  weeding  out  of 
these  invalid  and  short  methods  through  increasing  familiarity  with 
fractions. 


Sai  D-  rr*<iieat 


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CraJaL9 


Diagram  10. — Acomparisonof  the  median  numbers  of  "attempts  "and  "rights" 
for  the  several  grades  in  Set  O  (fractions)  and  Set  M  (addition). 


At  this  point  in  the  discussion  Diagram  10  may  be  introduced, 
although  it  is  not  a  diagram  of  distributions.  But,  since  it  bears 
out  what  has  just  been  said,  it  will  do  no  harm  to  examine  it.  There 
are  two  sections  in  the  diagram,  the  one  showing  curves  of  median 
"attempts"  and  median  "rights"  through  the  grades  for  Set  O, 
fractions;  the  other  doing  the  same  thing  for  Set  M,  addition.  The 
striking  fact  brought  out  by  the  diagram  is  that  the  median  number 
of  examples  attempted  decreases  from  Grade  7-2,  while  the  median 


GENERAL  RESULTS  47 

of  "rights"  either  increases  or  remains  stationary.  As  pointed  out 
in  the  previous  paragraph,  this  indicates  the  substitution  of  an  easy 
invalid  method  for  the  valid  one.  All  of  this  goes  to  show  that  the 
teaching  of  fractions  is  not  a  simple  matter  and  should  not  be  con- 
fused with  the  teaching  of  the  fundamentals. 

ACCURACY 

Although  the  importance  of  standards  of  accuracy  is  clearly 
recognized,  only  a  very  brief  study  has  been  made  of  accuracy 
because  of  the  immense  amount  of  labor  involved  in  the  under- 
taking. However,  a  study  has  been  made  of  2,400  cases,  400  from 
each  grade  taken  at  random  from  the  records  made  by  the  Cleve- 
land children.  The  records  were  all  scored  by  the  writer,  lest  an 
error  enter  into  the  results  due  to  the  scoring  by  the  pupils.  The 
results  of  this  study  are  found  in  Table  XVI.  But,  since  accuracy 
by  itself,  separated  from  a  statement  of  the  number  of  examples 
attempted,  has  but  little  meaning,  this  table  is  accompanied  by 
Table  XVII.  In  the  latter  appear  the  average  number  of  examples 
attempted  and  the  average  number  worked  correctly  by  this  same 
group  of  2,400  pupils  in  each  of  the  sets. 

Turning  now  to  Table  XVI,  it  is  noted  that  of  the  4  sets  in  the 
simple  combinations  (A,  B,  C,  D)  Set  C  seems  to  be  markedly 
the  most  difficult.  It  is  also  seen  that  from  the  third  grade  to  the 
eighth,  not  only  is  there  no  increase  in  accuracy  in  this  set,  but  there 
is  an  actual  decrease.  This  is  due  to  certain  peculiar  types  of  errors 
that  the  children  make  in  this  set.  A  complete  discussion  of  these 
errors  will  be  found  in  the  following  chapter.  Incidentally  this  in- 
accuracy throws  some  light  on  a  related  matter.  Reference  to  the 
standard  score  in  Table  II  shows  that  the  score  for  Set  C  is  smaller 
than  the  score  for  Set  D  in  the  upper  grades.  It  has  been  con- 
tended by  some  that  this  difference  is  to  be  accounted  for  by  the 
fact  that  in  writing  the  product  in  Set  C  two  figures  are  required 
in  most  cases,  while  in  Set  D  the  quotients  are  all  single  figures. 
Now  it  is  undoubtedly  the  case  that  this  is  a  factor,  but  that  it  is 
not  the  only  one  is  clearly  shown  by  this  table  on  accuracy  in  con- 
junction with  the  accompanying  table  on  "rights"  and  "attempts," 
Table  XVII.    The  latter  table  shows  that  the  difference  in  scores 


48 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


made  in  the  two  sets  is  largely  due  to  inaccuracy,  since  there  is  but 
little  difference  in  the  number  of  examples  attempted. 

A  further  examination  of  Table  XVI  shows  the  greatest  inaccu- 
racy to  be  found  in  fractions  (both  sets),  short  division,  the  addition 


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Diagram  ii. — A  comparison  of  percentages  of  accuracy  made  in  each  of  the 
fifteen  sefs  by  eighth-grade  pupils  in  Cleveland  and  Grand  Rapids. 

of  long  columns,  the  addition  involving  "carrying,"  and  the  multi- 
plication of  four-  by  two-place  numbers.  The  accuracy  is  quite 
high  in  long  division,  Sets  K  and  N.    This  is  probably  due  to  the 


GENERAL  RESULTS 


49 


frequent  change  in  the  type  of  mental  operation  demanded.     Thus 
long  mental  strain  is  avoided. 

In  Diagram  ii  the  accuracy  achieved  in  the  various  sets  by 
Cleveland  and  Grand  Rapids  eighth-grade  pupils  is  shown.  Cleve- 
land is  represented  by  400  cases  and  Grand  Rapids  by  150,  taken 
at  random.  In  general,  the  results  from  the  two  cities  agree  closely, 
the  two  more  obvious  exceptions  being  found  in  Sets  C  and  N.  In 
the  former  Grand  Rapids  is  quite  superior  to  Cleveland,  while 
in  the  latter  the  reverse  is  true.  It  should  be  remembered  in  this 
connection  that  earlier  in  the  chapter  it  was  found  that  in  median 
scores  the  weakness  of  Grand  Rapids  was  found  to  be  in  long 
division.  Thus  it  is  seen  that  this  weakness  is  further  indicated 
by  inaccuracy. 

TABLE  XVI 

Percentage  of  AcojRAcy  in  Each  Set  for  Grades  3-8.    Data  from  2,400  Pupils 


Set 


Grade 


A 

95-6 
91.9 
89.8 

B 

C 

D 

83.8 
87.2 

E 

F 

56.9 
61.4 
46.9 
28.0 

G 

H 

I 

J 

■54 -8 

K 

31-8 

L 

M 

45-5 

N 

0 

98.5 

96.3 
90.8 

93-4 
91.0 

76.1 
81.3 
74- 1 
46.6 

673 

78.8 
48.5 
63 -7 
29.1 
16.4 


98 
98 
90 

95 
92 

87 
8S 
68 
68 
73 


98.1 
88.4 
97.0 
93  5 

88.2 
86.9 
68.6 
75-8 
76.8 

90.3 
62.8 
69.9 
S8.6 
52.6 


98.2 

87.4 
97.2 

93  o 


873 
85-7 
735 
80.3 
75-8 


92.0 
62.  s 

73  o 
65.1 
S8.o 


98.9 
88.6 
97-3 
94-3 

90.4 
88.4 
76.9 
84.2 
78.0 

95-2 
68.9 

75-7 
81.0 
67.6 


The  accuracy  from  grade  to  grade  in  13  of  the  sets,  as  the  facts 
were  presented  in  Table  XVI,  is  shown  in  Diagram  12.  As  would 
be  expected,  the  diagram  shows  most  progress  to  be  made  in  the 
more  complex  types  of  examples.  In  the  simple  combinations 
there  is  very  little  progress.  As  a  general  rule  there  is  consistent 
progress  from  grade  to  grade.  To  this  statement  the  sets  in  frac- 
tions offer  exceptions,  as  they  do  in  a  great  many  respects. 


so 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


The  reader  has  perhaps  already  noted  the  fact  that  the  accuracy- 
curve  is  quite  diflferent  in  form  from  the  curve  of  "rights."    In 


A  (/(/it /on 


Muiiiptica  i'lO  n 


Srae/e. 


Diagram  i2. — Showing  the  progress  in  accuracy  in  each  of  thirteen  sets  through 
the  grades. 

Diagram  13  these  two  curves  are  compared  for  Set  K.    The  curve 
for  the  "rights"  possesses  that  quality  so  characteristic  of  the 


GENERAL  RESULTS 


51 


Courtis  standards — uniform  progress  from  one  grade  to  the  next. 
The  accuracy-curve,  on  the  other  hand,  resembles  the  learning- 
curve,  going  up  very  rapidly  at  first  and  gradually  turning  over  to 


TABLE  XVn 

Average  Rights  and  Average  Attempts  Made  in  Each  Set. 
Grades  3-8,  2,400  PtrpiLS 


Grade 

Set 

3 

4 

s 

6 

7 

8 

2 

Q. 
< 

on 

0. 

i 

< 

3 

% 
« 

1 

s 

< 

1 

a 

1 
< 

a 
< 

2 

1 

< 

A 

B 

C 

D 

E 

F 

G 

H 

I 

J 

K 

L 

16.3 
9.9 
7.2 
6.7 
4-4 

2.1 
1.6 
0.8 
0.4 
1-7 

0.02 

17.0 

10.8 
8.1 
8.0 
51 

3-7 

2-5 

1-7 
1-5 
30 

0.06 

0.1 

2.1 

20.3 
14.0 
13-5 
131 
5-7 

4.6 
35 
2.6 

I.O 

3-1 

2.7 
1-4 
2.4 
0.4 
0.1 

20.6 

14-5 

14.9 

14.0 

6.2 

6.1 
4-3 
3-5 
2.1 
4.6 

4-7 
30 
3.8 
1-3 
0.4 

233 
18.3 
iS-7 
16.6 
6.2 

6.9 

4.7 
4.2 
1.9 
3-7 

6.1 
2.1 
2.9 
0.9 
0.2 

23.6 
18.7 
17.4 
174 
6.7 

7.8 

5-5 
6.1 
2.8 
SO 

7.0 
3-6 
4-3 
1-7 
1.2 

25.2 
20.1 
17-4 
193 
6.5 

7-5 

S-2 

6.4 

2.8 

4.3 

8.2 
2.4 

35 
1.2 

3-9 

25 -5 
20.5 
ig.6 
19.9 
7.0 

8.5 
6.0 

9-3 
3-6 
5-6 

9.1 
3-8 
50 
2.0 
7-5 

28.1 

22.4 

18.7 

21.8 

7-5 

8.5 
S.8 
8.1 
3-6 
4.8 

10. 1 

2.8 
4.2 
1.6 
4.7 

28.4 
22.8 
21.4 
22.4 

8.0 

9.8 
6.7 
II. 0 
4-5 
6.3 

II. 0 

4-5 
5-8 
2.4 
8.2 

28.8 
25.8 
19.0 

22.8 

7-7 

9.6 
6.2 
8  8 
4-3 
5-5 

12.0 
3-4 
4-7 
2.2 
6.0 

29.1 
26.0 
21.5 
23-5 

8.2 

10.7 
7.0 

II. 4 
51 
71 

12.6 
50 
6.2 
2.7 
8.8 

M 

N 

0.9 

0 

the  horizontal.  From  this  comparison  it  would  seem  that  accuracy 
is  attained  as  the  result  of  practice,  thus  approaching  the  learning- 
curve,  while  the  development  of  speed  is  dependent  on  the  maturing 
of  the  pupil. 

SUMMARY 

I.  Standard  scores  for  the  several  sets  in  Grades  3-8  have  been 
determined  on  the  basis  of  results  secured  from  Cleveland  and 
Grand  Rapids  pupils.  A  comparison  of  these  scores  with  the 
Courtis  standard  scores  in  sets  A,  B,  C,  and  D  indicates  that  the 
scores  in  these  sets  constitute  quite  accurate  standards  of  attain- 
ment; and  there  seems  to  be  no  reason  for  believing  that  the  scores 


52 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


in  the  other  Sets,  with  the  possible  exception  of  set  H,  do  not  con- 
stitute equally  accurate  standards. 

2.  A  system  of  weights  has  been  derived  whereby  it  is  possible 
to  equate  the  scores  made  in  the  several  sets  so  that  a  single  score 

may  be  secured  to  represent 
the  general  arithmetical  at- 
tainment of  an  individual  or 
group. 

3.  The  use  of  the  test  is 
considered  at  some  length. 
Methods  of  diagnosing  indi- 
vidual, class,  school,  and 
city  weaknesses  are  indi- 
cated. 

4.  Some  very  interesting 
facts  are  brought  out  in 
comparing  grade  distribu- 
tions in  the  various  types  of 
examples.  First,  in  the  fun- 
damentals the  distribution- 
curve  tends  to  become 
flattened  with  progress 

through  the  grades.  Secondly,  the  distribution-curve  also  tends 
to  become  flattened  as  we  proceed  from  the  less  complex  to 
the  more  complex  types  of  examples  in  the  fundamentals.  Thirdly, 
as  a  general  proposition  in  the  fundamentals  the  distribution-curve 
representing  the  "rights"  is  flatter  than  that  representing  the 
"attempts."  .  Fourthly,  in  Set  O,  fractions,  the  exact  reverse  of 
this  last  statement  is  true,  the  curve  for  the  "attempts"  being 
flatter  than  that  for  the  "rights." 

5.  Tentative  standards  of  accuracy  for  each  of  the  sets  in 
Grades  3-8  inclusive  have  been  determined  on  the  basis  of  results 
from  Cleveland  and  Grand  Rapids  children. 

6.  Curves  representing  progress  in  accuracy  through  the  grades 
and  curves  representing  progress  in  the  average  number  of  examples 
worked  are  compared.  The  accuracy-curve  takes  the  form  of  the 
learning-curve,  while  the  "rights "-curve  does  not. 


Diagram  13. — ^A  comparison  of  curves  of 
acCTiracy  and  "rights"  for  Set  K  (long  division). 


CHAPTER  IV 
TYPES  OF  ERRORS 

This  study  represents  an  attempt  to  discover  the  different  types 
of  errors  made  in  the  various  sets  by  pupils.  In  order  to  keep  the 
study  within  limits  it  has  been  confined  ahnost  entirely  to  the 
eighth  grade,  a  few  comparisons  being  made  between  the  eighth  and 
fifth  grades  in  the  simple  combinations.  Furthermore,  it  has  been 
found  impossible  to  make  a  study  of  the  errors  made  in  certain  of 
the  sets,  because  of  the  impossibility  of  isolating  them.  For 
instance,  it  is  impossible  to  determine  beyond  a  reasonable  doubt 
from  the  record  made  by  a  pupil  in  working  an  example  in  Set  M, 
addition  of  5  four-place  numbers,  whether  or  not  an  error  in  the 
sum  is  due  to  a  mistake  made  in  adding  or  "carrying."  For  like 
reasons  no  attempt  is  made  to  study  errors  made  in  Sets  E,  F,  G, 
I,  and  J.  It  is  therefore  apparent  that  this  study  should  be  supple- 
mented by  experimentation. 

ADDITION 

For  the  reason  just  stated  Set  A  is  the  only  set  in  addition  that 
can  be  profitably  studied.  Certain  facts  relating  to  errors  made  in 
this  set  appear  in  Tables  XVllI  and  XIX  and  in  Diagram  14.  In 
the  first  table  there  is  a  comparison  of  the  distributions  of  100  errors 
made  in  the  first  26  simple  addition  combinations  of  the  set  by 
Cleveland  and  Grand  Rapids  eighth-grade  pupils.  It  will  be  noted 
that  these  26  combinations  include  the  first  two  rows  of  examples. 
In  order  to  determine  the  distribution  of  100  errors  for  these  com- 
binations, only  those  records  of  the  eighth-grade  pupils  for  each  city 
were  selected  in  which  the  pupils  had  attempted  all  the  examples 
in  the  first  two  rows.  That  is,  no  record  was  used  which  showed 
that  the  pupil  had  not  attempted  every  one  of  these  26  examples. 
This  method  makes  the  numbers  of  errors  for  the  26  combinations 
strictly  comparable.  Thus,  turning  to  the  table,  it  is  understood 
if  read  as  follows:  Of  the  100  errors  made  by  the  Cleveland  eighth- 
grade  pupils  on  the  26  combinations,  all  of  which  were  attempted 

S3 


54 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


the  same  number  of  times,  none  was  made  on  the  first  combination, 
1+2;  six  were  made  on  the  second,  6+6,  and  so  on.  In  the  second 
table  the  fifth  and  eighth  grades  of  Grand  Rapids  are  compared  in 
the  same  way  as  were  the  two  eighth  grades  in  the  first  table,  with 
the  exception  that  the  100  errors  are  distributed  over  but  15  com- 
binations. In  Diagram  14  are  reproduced  certain  interesting  and 
typical  errors  actually  made  by  the  pupils. 

An  examination  of  the  combined  results  from  Cleveland  and 
Grand  Rapids  in  Table  XVIII  shows  the  easiest  combinations  to 
be  1+2,  7+7,  0+7,  and  3+1.  With  the  exception  of  the  second 
of  these  combinations,  the  sum  is  in  each  case  less  than  ten.     For 


TABLE  XVIII 

A  Comparison  of  the  Distribxttions  of  100  Errors  Made  in  26  Simple  Addition  Combi- 
nations BY  Eighth-Grade  Pupils  in  Cleveland  and  Grand  Rapids 


Simple  Addition  Combinations 

City 

I 
3 

6 
6 

9 
S 

0 

I 

4 

3 

I 
3 

7 
7 

9 
6 

3 
0 

3 
4 

I 
5 

3 
8 

6 
9 

0 
7 

3 

3 

8 

I 

9 
9 

7 
6 

8 
0 

3 
S 

I 

6 

4 
7 

8 
9 

0 

S 

3 

7 

3 

I 

Total 

6 

S 

9 

4 

3 
3 

3 
3 

I 
I 

6 
9 

3 

4 

4 
17 

S 
7 

4 
II 

I 

8 
12 

3 

1 

3 
S 

7 
4 

I 
I 

I 
1 

5 
4 

6 
3 

7 
3 

I 
I 

II 
8 

I 

100 

Grand  Rapids 

Total. . . 

IX 

13 

4 

4 

3 

IS 

3 

4 

31 

12 

IS 

I 

20 

3 

8 

II 

3 

3 

9 

9 

9 

3 

19 

I 

300 

*• 

some  reason  the  association,  7 + 7  =  14,  is  very  strong.  There  is  even 
a  tendency,  as  will  be  pointed  out  in  the  discussion  of  Set  C,  to  add 
7  and  7  when  the  combination  appears  in  a  set  of  examples  in 
multiplication.  A  further  curious  fact  is  that  the  three  combinations 
1+5,  3+2,  and  2+7  are  the  most  difficult,  or  rather  net  the  most 
errors,  and  yet  their  sums  are  also  less  than  ten  in  each  case.  And 
there  seems  to  be  a  fixed  association  between  each  of  these  and  a 
sum  which  is  incorrect.  The  t3^ical  associations  for  these  com- 
binations when  errors  are  made  appear  in  sections  b,  c,  and  d  of 
Diagram  14.  There  is  a  strong  tendency  to  say  3+2  =  6,  1+5  =  7, 
and  2+7  =  8.  These  three  errors  account  for  practically  all  the 
errors  made  in  these  combinations.  However,  with  these  excep- 
tions errors  are  made  more  frequently  with  the  larger  than  with  the 
smaller  combinations.  Of  the  200  total  errors  considered,  the  ten 
combinations  whose  sums  are  greater  than  ten  show  the  average 


TYPES  OF  ERRORS  55 

number  of  errors  made  per  combination  to  be  10.3,  while  the  aver- 
age number  for  the  sixteen  combinations  with  sums  less  than  ten  is 
but  6.1.  Now,  if  the  combination  7+7  be  eliminated  from  the 
first  group,  and  the  combinations  5+r,  3+2,  and  7+2  be  eliminated 
from  the  second  group,  the  difference  is  much  more  striking,  being 
an  average  of  11. 4  errors  for  the  former  and  4.6  for  the  latter. 
Thus  it  is  seen  that  with  exceptions  the  larger  combinations  are  the 
more  difficult,  or  at  least  the  associations  between  them  and  their 
sums  are  weaker,  than  the  smaller  combinations. 

96  3  I  2 

-^        _?  _^  J  _7 

17        17  6  7  8 

a  bed 

Diagram  14. — Typical  errors  made  in  Set  A,  simple  addition 

The  comparison  between  Cleveland  and  Grand  Rapids  shows 
certain  differences.  The  errors  are  more  evenly  distributed  over 
the  26  combinations  for  the  former  than  for  the  latter,  the  largest 
number  of  errors  made  on  any  one  combination  by  the  Cleveland 
pupils  being  11  as  opposed  to  17  for  the  pupils  of  Grand  Rapids. 
It  is  also  true  that,  while  on  the  whole  there  is  rather  close  agree- 
ment as  to  the  difficulty  of  the  several  combinations,  there  are 
several  quite  marked  exceptions  to  this  statement.  The  1+5  and 
3-f-2  combinations  net  more  errors  in  Grand  Rapids  than  in  Cleve- 
land, while  the  reverse  is  true  for  2+7,  9+5,  and  several  others. 
This  would  indicate  that  certain  rather  freakish  associations  are 
established  in  different  groups  through  peculiar  methods  of  instruc- 
tion or  some  other  experience  common  to  the  individuals  making 
up  each  of  the  groups. 

This  last  statement  seems  to  be  borne  out  by  the  facts  presented 
in  Table  XIX,  in  which  the  fifth  and  eighth  grades  are  compared. 
The  association  1+5  =  7,  so  strongly  established  in  the  eighth-grade 
pupils  of  Grand  Rapids,  was  found  to  be  relatively  weak  for  the 
Cleveland  group,  and  is  here  seen  to  be  quite  weak  for  the  fifth 
grade  in  Grand  Rapids.  The  association  3+2  =  6  is  strong  in  both 
grades.  In  general  the  statement  made  concerning  the  relative 
difficulties  of  the  combinations  for  the  eighth  grade  holds  for  the  fifth. 


S6 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


An  additional  comment  may  be  made  with  reference  to  the 

formation  of  wrong  associations.    An  examination  of  the  raw  data 

shows  in  numerous  instances  a  rather  strong  persistence  of  these 

wrong  associations.     In  addition  to  those  aheady  mentioned  there 

appears  another  in  section  a  of  Diagram  14.     These  two  errors, 

9+6  =  17  and  6+9=17,  were  taken  from  the  same  record.    The 

repetition  of  the  error  would  indicate  that  the  association  was  quite 

strongly  fixed  and  in  all  probability  denotes  a  confusion  between 

9+6  and  9+8. 

TABLE  xrx 

A  Comparison  of  the  Distributions  of  100  Errors  Made  in  15  Simple  Addition 
Combinations  by  Fifth-  and  Eighth-Grade  Pupils  in  Grand  Rapids 


Simple  Addition  Combinations 

Grade 

I 
a 

6 
6 

9 
s 

0 

I 

4 

2 

X 

3 

7 
7 

9 
6 

3 
0 

3 

4 

s 

i 

6 
9 

0 
7 

3 

3 

Total 

8-2 

7 

I 

6 
10 

3 
6 

3 

5 

I 

4 

I 

13 

8 

3 

5 

24 
4 

10 
II 

16 
II 

6' 

17 
25 

100 

C-2 

100 

Total 

8 

16 

9 

8 

5 

I 

21 

3 

5 

28 

21 

27 

6 

42 

200 

SUBTRACTION 

/The  study  of  errors  made  in  subtraction  is  confined  to  Set  B. 
Facts  corresponding  in  essential  features  to  those  presented  on 
addition  are  found  in  Tables  XX  and  XXI  and  Diagram  15,  bearing 
on  subtraction.  The  one  important  difference  is  that  but  20  com- 
binations are  studied  in  the  eighth  grade  and  10  in  the  fifth. 

From  the  totals  in  Table  XX  it  is  seen  that  bridging  the  tens  is 
a  relatively  much  more  difficult  operation  in  subtraction  than  in 
addition.  In  the  latter  it  was  found  that  the  average  number  of 
errors  made  in  the  combinations  whose  sums  were  greater  than 
ten,  excluding  certain  rather  freakish  results,  was  11 .9  as  opposed 
to  4.6  for  the  combinations  whose  sums  were  less  than  ten.  With- 
out making  any  exception  it  is  found  in  subtraction  that  the  average 
number  of  errors  made  where  the  minuend  is  more  than  ten  is  18.7 
(nine  cases),  while  the  average  where  the  minuend  is  ten  or  less 
is  but  2.9  (eleven  cases). 

In  the  comparison  of  the  fifth  and  eighth  grades  in  Table  XXI 
one  very  interesting  difference  is  noted.    Whereas  the  eighth-grade 


TYPES  OF  ERRORS 


57 


pupils  have  relatively  little  difficulty  with  the  combination  i— o, 
it  is  by  far  the  most  difficult  combination  for  the  fifth  grade, 
accounting  for  40  errors  out  of  the  100.  As  will  appear  in  the  dis- 
cussion of  multiplication,  the  pupil  either  has  a  great  deal  of  diffi- 
culty in  getting  the  conception  of  zero,  or  practically  no  attention  is 
given  to  it  in  the  course  of  instruction  in  arithmetic.  From  the 
facts  presented  in  this  table  it  would  seem  that  the  understanding 
of  what  zero  means  accompanies  the  maturing  of  the  pupil.  With 
this  one  exception  differences  between  the  two  grades  are  not  par- 
ticularly evident. 

TABLE  XX 

A   COJCPARISON  or  THE  DISTRIBUTIONS  OF   lOO  ERRORS  MaDE  IN   20  SiMPLE   SUB- 
TRACTION Combinations  by  Eighth-Grade  Pupils  in  Cleveland 
AND  Grand  Rapids 


Simple  Subtraction  Combinations 

City 

9 
9 

7 
3 

II 
6 

8 

I 

12 
3 

I 
0 

9 

7 

13 
8 

4 
3 

12 
6 

8 
0 

II 
9 

13 

7 

s 

I 

10 

3 

6 
0 

II 

7 

IS 

10 
9 

13 

s 

Total 

Cleveland 

2 

I 

8 
4 

2 

3 

14 

7 

S 

I 
2 

II 
II 

I 

I 
3 

I 

10 

s 

7 
9 

I 
4 

2 

I 
2 

18 
18 

12 
20 

3 

5 
S 

100 

Grand  Rapids 

I 

ICX) 

Total 

I 

3 

12 

5 

21 

5 

3 

22 

I 

4 

I 

IS 

16 

5 

2 

3 

36 

32 

3 

10 

200 

TABLE  XXI 

A  Comparison  of  the  Distributions  of  100  Errors  Made  in  10  Simple  Sub- 
traction Combinations  by  Fifth-  and  Eighth-Grade  Pupils  in 
Grand  Rapids 


Simple  Subtraction  Combinations 

Grade 

9 
9 

7 
3 

II 
6 

8 

I 

13 
3 

I 
0 

9 
7 

13 
8 

4 
3 

13 

6 

Total 

8-2 

2 
3 

3 

5 

II 
8 

8 

19 
14 

14 
40 

S 
5 

30 
23 

2 

8 

ICO 

C-2 

ICO 

Total 

5 

8 

19 

8 

33 

54 

10 

Si 

2 

8 

200 

In  Diagram  15  there  are  presented  two  t3^ical  errors.  The  error 
in  section  h  of  the  diagram,  1—0=0,  characteristic  of  the  fifth 
grade,  has  already  been  commented  upon.  The  two  errors  in  sec- 
tion a,  11—7  =  5  3.nd  12  —  7=4,  were  made  by  the  same  pupil. 


58 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


They  indicate  the  fixing  of  the  wrong  associations  referred  to  in 
connection  with  the  addition  combinations.  It  is  evident  that  the 
halves  of  two  associations  were  wrongly  paired. 


II 

_7 

S 


la 
_7 

4 


Diagram  15. — ^Typical  errors  made  in  Set  B,  subtraction 
MULTIPLICATION 

In  studying  errors  in  multiplication  two  sets  were  used,  C  and  L. 
The  errors  characteristic  of  these  two  types  will  be  discussed  in  their 
order. 

In  form,  Tables  XXII  and  XXIII  are  identical  with  the  corre- 
sponding tables  for  addition  and  subtraction.  In  the  first  the 
eighth  grades  of  Cleveland  and  Grand  Rapids  are  compared,  in  the 
second,  the  fifth  and  eighth  grades  of  the  latter  city.  Tjqjical 
errors,  as  actually  made  by  the  pupils,  are  reproduced  in  Dia- 
gram 16. 

Turning  to  Table  XXII,  we  note  a  striking  difference  between 
th§  distribution  of  errors  in  multiplication  and  the  distribution  of 
errors  in  addition  and  subtraction,  already  discussed,  and  in  the 


TABLE  XXII 

A  Comparison  op  the  Distributions  of  100  Errors  Made  in  20  Simple  Multi- 
plication Combinations  by  Eighth- Grade  Pupils  in  Cleveland  and 
Grand  Rapids 


Simple  Multiplication  Combinations 

,           Gty 

3 

2 

4 

7 

i 

0 
2 

I 

4 

I 

2 
9 

7 
6 

4 
0 

9 
5 

9 

I 

s 

2 

8^ 

7 
0 

6 
S 

2 
I 

3 
3 

9 
6 

0 

s 

7 

4 

Total 

Cleveland 

24 
17 

2 

26 
28 

I 

I 

26 
28 

1 

I 

2 

2 

I 

19 
18 

I 

100 

Grand  Rapids 

2 

100 

Total 

2 

41 

2 

54 

I 

I 

54 

2 

2 

3 

37 

I 

200 

concentration  of  errors  at  certain  points.  It  is  in  those  combina- 
tions into  which  zero  enters  as  one  of  the  terms  that  the  largest 
number  of  errors  is  made.    This  statement  is  equally  true  of  both 


TYPES  OF  ERRORS 


59 


Cleveland  and  Grand  Rapids,  as  well  as  of  the  fifth  and  eighth 
grades.  The  similarity  of  the  results  from  Cleveland  and  Grand 
Rapids  is  of  especial  interest  and  significance  when  it  is  remembered 
that  the  children  of  the  latter  city  were  familiar  with  the  Courtis 
tests.  Thus,  in  spite  of  whatever  special  training  they  may  have 
received  from  these  tests  on  the  zero  combinations,  they  registered 
about  the  same  proportion  of  errors  on  these  combinations  as  did 
the  Cleveland  children  who  had  not  had  this  special  training.  Thus 
it  would  seem  that  the  handhng  of  the  zero  is  a  mental  function 
peculiarly  unresponsive  to  training. 


TABLE  XXIII 

A  Comparison  of  the  Distributions  of  ioo  Errors  Made  in  io  Simple  Multi- 
plication Combinations  by  Fifth-  and  Eighth-Grade  Pupils  in 
Grand  Rapids 


Simple  Multiplication  Combinations 

Grade 

3 

2 

4 
7 

9 
8 

o 

2 

5 
6 

4 

I 

3 

9 

7 
6 

4 
o 

9 

S 

Total 

8-2 

4 

I 

"k" 

35 

25 

3 

4 

57 
55 

3 

lOO 

c-2 

I 

I 

3 

IOO 

Total 

I 

5 

8 

6o 

3 

4 

I 

3 

1X2 

3 

200 

A  further  study  of  the  zero  as  it  enters  into  the  various  combina- 
tions is  interesting.  The  reader  has  probably  already  noticed  the 
greater  frequency  of  errors  at  0X4  and  0X7  than  at  2X0  and 
5X0.  Turning  now  to  Diagram  16,  sections  a,  b,  and  c,  we  find 
the  three  typical  performances  in  dealing  with  the  four  zero  com- 
binations when  an  error  is  made.  A  pupil  may  respond  correctly 
to  each  of  the  combinations,  he  may  fail  on  2X0  and  5X0,  he  may 
fail  on  0X4  and  0X7,  or  he  may  fail  on  all  four  combinations.  A 
striking  fact  suggested  by  the  tables  and  borne  out  completely  by 
an  examination  of  the  actual  work  of  the  children  is  that  the  making 
of  errors  on  these  four  combinations  goes  in  pairs.  That  is,  there 
is  a  tendency  to  fail  on  2X0  and  5X0,  while  giving  the  proper 
reaction  to  the  other  two  combinations,  0X4  and  0X7,  or  vice 
versa.  And,  as  seen  in  the  table,  the  error  is  more  frequently  made 
in  the  latter  pair  of  combinations  than  in  the  former.     Indeed,  it 


6o  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

is  very  rarely  the  case  that  a  pupil  fails  on  2X0  and  5X0  while 
reacting  properly  to  0X4  and  0X7.  Thus  it  would  seem  that  it 
is  a  more  difficult  mental  operation  to  multiply  a  quantity  by  zero 
than  to  perform  the  reverse  operation,  to  multiply  zero  by  the 
quantity.  It  is  further  evident  that  the  two  operations  are  not 
identical.  A  very  interesting  question  suggested  by  all  the  fore- 
going is  that  of  the  relation  between  dealing  with  zero  in  the  simple 
combinations  and  dealing  with  it  in  the  more  complex  multiplica- 
tion exaniples.  In  section  e  of  Diagram  16  we  have  reproduced  the 
work  of  a  pupil  who  was  quite  unable  to  handle  the  zero  in  the 


2 

0 

2              0 

2 

0 

3 

7_ 

2 

0 

0              4 

2 

4 

6 

r4 

7 

0 

7          0 

7 

0 

0 

_S 

0          5 

0 

_5 

0 

5 

a 

'    5     ° 

7 

c 

5 

d 

0 

4 

8563 

0 

4 

8563 

2 

0 

207 

'  2 

0 

207 

2 

4 

59941 
17126 

0 

0 

S994I 
171260 

7 
0 

0 
_5 

1772541 

7 
0 

0 
_5 

231201 

7 

S 
e 

0 

0 

/ 

Diagram 

16. — ^Typical  errors 

made 

in 

multiplication 

simple  combinations,  yet  had  not  the  least  difficulty  with  the  zero 
when  it  appeared  in  a  complex  example.  In  section/,  on  the  other 
hand,  there  is  reproduced  the  work  of  a  pupil  who  had  difficulty  in 
the  reverse  order,  handling  all  the  zero  combinations  correctly,  but 
failing  on  the  zero  in  the  complex  example.  It  should  also  be  said 
that  in  not  a  single  case  out  of  50  examined  did  the  writer  find 
difficulty  with  the  former  to  be  correlated  with  difficulty  with  the 
latter. 

One  more  type  of  error  should  be  mentioned  before  we  leave 
multiplication  combinations,  and  that  is  the  error  reproduced  in 
section  d  of  the  diagram,  which  is  mentioned  earUer  in  the  chapter 
in  connection  with  the  discussion  of  addition.  The  two  errors  re- 
produced, made  by  the  same  individual,  are  3X3  =  6  and  7X7  =  14. 


TYPES  OF  ERRORS  6l 

These  are  the  typical  errors  made  on  these  combinations. 
When  we  consider  that  these  responses  were  made  to  the  combina- 
tions when  they  appeared  in  the  midst  of  a  set  of  multipUcation 
examples,  when  the  pupils  were  actually  in  the  multiplying  atti- 
tude, the  evidence  points  to  a  relatively  strong  additional  associa- 
tion with  these  combinations.  Whether  or  not  there  is  a  tendency 
to  add  like  quantities  is  a  problem  demanding  further  experimental 
evidence  for  its  solution. 

Let  us  proceed  now  to  the  study  of  the  errors  made  in  the 
examples  of  Set  L,  the  multiplication  of  four-place  by  two-place 
niunbers.  The  facts,  as  they  have  been  secured,  appear  in  Table 
XXIV.  In  this  table  is  found  the  distribution  of  loo  errors  made 
by  the  eighth-grade  pupils.  It  will  be  noted  that  the  errors  are 
thrown  into  three  categories,  "mistakes  in  multiplying,"  "mistakes 

TABLE  XXIV 

Distribution  of  ioo  Errors  Made  in  Complex  Mul- 
tiplication Examples,  Set  L — Eighth  Grade 


Mistakes  in  multiplying. 
Mistakes  in  adding  .... 
Other  mistakes 


Total. 


72 

25 

3 


in  adding,"  and  "other  mistakes."  It  is  to  be  regretted  that  a 
more  detailed  study  might  not  have  been  made  of  the  errors  in 
these  examples,  but  this  was  found  to  be  impossible  from  the  mere 
records  of  the  pupils.  It  would  have  been  desirable  to  isolate  errors 
caused  through  "carrying"  in  performing  the  multipUcation  and 
"carrying"  in  performing  the  addition  necessary  to  the  solution 
of  this  type  of  example,  but  it  was  impossible  to  determine  whether 
a  mistake  that  appeared  in  one  of  the  partial  products  was  due  to 
difficulty  with  the  tables  or  with  "carrying."  Thus  "mistakes  in 
multiplying"  is  a  composite  including  both  mistakes  in  the  tables 
and  mistakes  in  "carrying,"  as  well  as  mistakes  due  to  the  combina- 
tion of  these  two  operations.  "Mistakes  in  adding"  is  likewise  a 
composite.  "Other  mistakes"  includes  mistakes  due  to  crowding 
of  figures,  difiiculty  with  the  mechanics  of  multiplication,  etc. 


62 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


A  word  concerning  the  table  itself  will  sufl&ce.  Mistakes  in 
multiplying  are  by  far  the  most  frequent.  This  merely  confirms 
facts  brought  out  in  the  tables  on  accuracy  which  showed  that  addi- 
tion is  always  performed  with  relatively  greater  accuracy  than  mul- 
tiplication. 

DIVISION 

In  the  study  of  typical  errors  in  division,  the  study  is  limited  to 
Sets  D  and  N;  Set  I  is  eliminated  because  of  the  impossibility  of 
isolating  mistakes;  and  the  facts  for  Set  K  are  not  presented  because 
it  IS  of  the  same  general  character  as  Set  N. 

The  results  of  the  study  of  Set  D  are  found  in  Tables  XXV  and 
XXVI  and  Diagram  17.  In  essential  respects  the  tables  are  like 
the  tables  for  the  three  other  sets  of  the  simple  combinations  and 

TABLE  XXV 

A  Comparison  of  the  Distmbtjtions  of  100  Errors  Made  in  21  Simple  Division 
Combinations  by  Eighth-Grade  Pupils  in  Clevelajto  and  Grand  Rapids 


Simple  Division  Combinations 

City 

? 
3 

32 
4 

36 
6 

0 
a 

28 
7 

9 
9 

31 

3 

48 
6 

I 
I 

10 
s 

6 
a 

4 

63 

7 

0 
6 

32 
8 

8 
I 

'30 
s 

72 

8 

0 

I 

36 
9 

7 

I 

Total 

Cleveland 

I 

3 

4 
2 

2 

I 

2 

32 
26 

I 

2 

5 

34 
24 

3 

I 

I 

8 

2 
2 

5 

I 

I 
I 

2 

4 

2 
S 

3 
3 

2 

I 

5 
9 

Grand  Rapids 

Total 

4 

6 

3 

2 

58 

I 

7 

58 

3 

I 

9 

4 

6 

2 

6 

7 

6 

3 

14 

therefore  they  require  no  explanation.  Attention  is  directed,  how- 
ever, to  the  fact  that  the  results  for  21  combinations  instead  of  for 
20  are  presented  in  the  first  table. 

An  examination  of  the  tables  shows  that  the  most  frequent  error 
is  made  in  those  examples  in  which  a  quantity  is  divided  by  itself. 


o 


9)9 

o 


Diagram  17. — Typical  error  made  in  Set  D,  division 

The  nature  of  this  error  is  made  clear  by  reference  to  Diagram  17. 
Here  it  is  seen  that  there  is  a  tendency  for  the  child  to  say  i-v- 1  =  0, 
9-r  9= o.     This  is  the  typical  error  made  in  the  simple  division  com- 


TYPES  OF  ERRORS 


63 


binations.  It  is  difl&cult  for  the  child  to  know  that  there  is  any- 
thing left  when  a  quantity  is  divided  by  itself.  It  is  at  this  point 
that  division  and  subtraction  become  confused. 

The  records  from  Cleveland  and  Grand  Rapids  are  in  distinct 
agreement  in  showing  this  to  be  the  most  frequent  error.  Rela- 
tively, however,  it  seems  to  be  made  more  frequently  by  children 
from  the  former  than  by  those  from  the  latter  city.  Table  XXVI 
shows  this  error  to  stand  out  prominently  in  the  fifth  grade  as  well 
as  in  the  eighth.  The  fifth-grade  pupils,  however,  have  relatively 
more  difficulty  with  other  combinations  than  do  the  eighth-grade 
pupils.  This  is  just  what  should  be  expected  when  we  consider 
that  it  merely  indicates  a  more  complete  mastery  of  the  tables  on 
the  part  of  the  pupils  in  the  more  advanced  grade. 

TABLE  XXVI 

A  Comparison  of  the  Distributions  of  icxj  Errors  Made  in  10  Simple  Division 
Combinations  by  Fifth-  and  Eighth-Grade  Pupils  in  Grand  Rapids 


Grade 

Simple 

Divbion  Combinations 

9 

3 

32 

4 

36 

6 

0 

3 

28 
7 

? 
9 

31 

3 

48 

6 

I 
I 

10 

5 

Total 

8-? 

S 

s 

3 
13 

2 

4 

3 

S 

39 

25 

"e" 

8 
7 

36 
26 

4 
5 

100 

5-2 

4 

lOO 

Total 

4 

10 

16 

6 

8 

64 

6 

IS 

62 

9 

200 

In  studying  the  errors  made  in  examples  in  long  division.  Set  N, 
they  were  grouped  into  three  groups  corresponding  to  the  three 

TABLE  XXVII 

Distribution  of  100  Errors  Made  in  Examples  in 
Long  Division,  Set  N — Eighth-Grade  Pupils 


Mistakes  in  multiplying 
Mistakes  in  subtraction 
Other  mistakes 

Total 


68 

26 

6 


groups  employed  in  connection  with  the  multiplication.  Set  L. 
These  three  groups,  as  shown  in  Table  XXVII,  are  "mistakes  in 


64  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

multiplying,"  "mistakes  in  subtraction,"  and  "other  mistakes." 
An  explanation  of  these  terms  is  not  necessary.  Suffice  it  to  say 
that  each  of  the  groups  of  errors  has  the  same  composite  character 
as  was  referred  to  in  the  discussion  of  multiplication. 

The  table  shows  marked  similarity  to  Table  XXIV.  A  large 
number  of  mistakes  is  made  in  multiplying,  and  in  the  processes 
incident  thereto,  while  a  comparatively  small  number  is  made  in 
subtraction.  This  further  bears  out  the  proposition  that  multipli- 
cation is  a  relatively  difficult  mental  operation. 

FRACTIONS 

Fractions,  it  will  be  remembered,  are  represented  in  the  test  by 
two  sets,  H  and  O.  In  the  first,  fractions  of  like  denominators  are 
added  and  subtracted;  in  the  second,  fractions  of  unlike  denomina- 
tor are  added,  subtracted,  multiplied,  and  divided.  As  compared 
with  the  more  complex  examples  in  the  "fundamentals,"  it  has  been 
found  relatively  easy  to  make  a  study  of  errors.  From  the  result 
set  down  by  the  pupil  it  is  possible  in  most  cases  to  determine  what 
he  did  and  how  he  did  it.  For  this  reason  a  more  exhaustive  study 
has  been  undertaken  of  the  errors  made  in  these  two  sets  than  was 
possible  in  connection  with  any  one  of  the  four  fundamental  opera- 
tions. It  should  be  added  further  that,  since  each  of  these  sets  is 
a  complex,  a  separate  study  has  been  made  of  the  errors  occurring 
in  each  of  the  types  of  operation  found  in  the  sets.  Thus  the  study 
of  Set  H  is  divided  into  two  parts,  the  one  concerned  with  the  addi- 
tion, the  other  with  the  subtraction,  of  fractions  of  like  denomina- 
tors. The  study  of  Set  O  is  consequently  divided  into  four  parts, 
dealing  respectively  with  addition,  subtraction,  multiplication,  and 
division  of  fractions  of  unlike  denominators. 

FRACTIONS  OF  LIKE  DENOMINATORS 

One  hundred  errors  made  in  the  addition  of  fractions  of  like 
denominators  by  eighth-grade  pupils  were  analyzed  and  thrown  into 
the  five  categories  which  appear  in  Table  XXVIII.  In  dealing  with 
these  simple  fractions  the  most  frequent  error  is  found  to  be  that 
indicated  by  the  reproduction  in  section  a,  Diagram  i8.  The 
numerators  are  added  and  likewise  the  denominators.    This  is 


TYPES  OF  ERRORS 


6S 


perhaps  the  mistake  that  would  be  expected,  and  it  should  therefore 
be  very  carefully  guarded  against  by  the  teacher.  It  shows,  how- 
ever, that  the  child  does  not  have  the  least  conception  of  the 
meaning  of  denominator. 

TABLE  XXVIII 

Freqtiency  of  Types  of  Error  in  Addition  of  Fractions  of 
Like  DENoiiiNATORS — Eighth  Grade 


Type  of  Error 

Numerators  added,  denominators  added 

Numerators  multiplied,  denominators  multiplied 
Numerators  added,  denominators  multiplied .... 

Common  denominator  found 

Numerators  multiplied,  denominators  added 

Total 


Frequency 


60 

27 

8 

4 

I 


Another  frequent  error  is  that  shown  in  section  b  of  the  diagram; 
the  numerators  are  multiplied  and  likewise  the  denominators. 
This  indicates  an  interference  of  mental  functions.     Since  the  pupil 


5      5     10 

9     9    18 


5      5     25 

9    9  81 

b 


5      5^25 
9      9  '81 


^+' 


20 
S      5     25 


9      9     81 

d 


5      5     10 
9      9     18 


3.,  J.  ^4. 
5      S     10 

15     6     30 

/ 
Diagram  18. — Typical  errors  made  in  adding  fractions  of  like  denominators 


is  an  eighth-grade  pupil,  the  multiplication  of  fractions  is  more 
vividly  in  his  mind  than  is  addition.  He  consequently  multiplies. 
Sometimes  an  individual  will  be  found  who  multiplies  everything 


66  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

in  Sets  H  and  0.  This  may  be  due  to  carelessness  in  observing 
signs;  or  it  may  be  due  to  the  fixing  of  the  method  of  handling  one 
type  of  fractions  at  the  expense  of  others. 

Another  type  of  error  is  found  in  section  c  of  the  diagram.  The 
numerators  are  added  and  the  denominators  multiplied.  This  indi- 
cates a  confusion  between  the  method  of  adding  and  the  method 
of  multipling  fractions.  A  modification  of  this  same  type  of  con- 
fusion appears  in  section  e  where  the  numerators  are  multiplied 
and  the  denominators  added.  And  finally  in  section  d  we  have  a 
performance  which,  while  not  an  error,  strictly  speaking,  indicates 
a  slavish  adherence  to  the  mechanics  of  fractions.  The  pupil, 
instead  of  simply  adding  the  numerators,  first  found  a  common 
denominator  and  then  added  the  new  numerators.  The  result  is 
not  wrong,  but  the  method  used  to  get  it  is  wrong. 

In  the  last  section  of  the  diagram,/,  the  work  of  a  pupil  in  adding 
fractions  of  like  denominators  and  in  adding  fractions  of  unlike 
denominators  is  reproduced.  It  would  seem  that  if  a  pupil  could 
work  examples  of  the  more  complex  type  he  would  have  no  difficulty 
in  working  those  of  the  simple  type,  especially  when  it  is  borne  in 
mind  that  in  the  course  of  instruction  he  encounters  the  latter  before 
the  former,  and  it  is  supposed  by  all  that  mastery  of  the  simple 
necessarily  underlies,  and  leads  up  to,  the  more  complex  operation. 
From  this  section  of  the  diagram  it  is  evident  that  this  is  a  false 
assumption.  Unless  the  pupil  is  given  an  understanding  of  the 
nature  of  fractions,  he  becomes  a  slave  to  the  method;  and  the 
learning  of  the  method  of  handling  one  type  of  fractions  seems  in  no 
way  to  involve  the  learning  of  the  method  of  handling  a  simpler 
type  of  fractions,  although  the  understanding  of  the  former  does 
involve  the  understanding  of  the  latter. 

In  Table  XXIX  and  Diagram  19  there  appears  a  corresponding 
analysis  of  the  types  of  error  made  by  eighth-grade  pupils  in  the 
subtracting  of  fractions  of  like  denominators.  It  will  be  noted 
that  the  most  frequent  error  is  "confusion  of  symbols,"  while  this 
error  did  not  occur  at  all  in  the  addition  of  similar  fractions.  This 
is  due  to  an  unhappy  organization  of  the  set.  An  examination  of 
this  set  shows  it  to  be  composed  of  four  columns  of  examples.  All 
those  in  the  first  column  are  to  be  added,  all  in  the  second  sub- 


TYPES  OF  ERRORS  67 

tracted,  all  in  the  third  added,  and  all  in  the  fourth  subtracted. 
Now,  since  the  fractions  in  the  first  column  are  to  be  added,  the 
pupil  quite  frequently  obeys  the  suggestion  that  all  the  fractions 
in  the  set  are  to  be  added.  Thus  "confusion  of  symbols"  appears 
as  a  frequent  type  of  error  in  subtraction  and  is  absent  from  addi- 
tion. 

TABLE  XXDC 

Freqitency  or  Types  of  Error  in  Subtraction  of  Fractions 
OF  Like  Denominators — Eighth  Grade 


Type  of  Error 


Confusion  of  symbols 

Numerators  subtracted,  denominators  subtracted. 
Numerators  multiplied,  denominators  multiplied. 

Numerators  added,  denominators  multiplied 

Numerators  subtracted,  denominators  added. ... 

Total 


Frequency 


43 

25 

23 

6 

3 


The  next  most  frequent  error  corresponds  to  the  most  frequent 
error  in  addition,  the  numerators  being  subtracted  and  the  denomi- 
nators subtracted.  The  exact  character  of  this  error  is  made  clear 
through  section  a  of  Diagram  19.  Another  error  appearing  with 
about  the  same  frequency  in  subtraction  as  in  addition  is  the  mul- 
tiplication of  both  numerators  and  denominators.     A  third  mistake, 


6      4  _  2 
9      9~o 

9      9     81 

6      4  _io 
9      981 

6      4  _  2 

9      9~i8 

770 

a 

7      7     49 

h 

7      7     49 

c 

7~7     14 

i 

Diagram  19. — Typical  errors  made  in  subtracting  fractions  of  like  denominators 

the  exact  nature  of  which  is  perhaps  due  to  confusion  of  signs,  but 
which  would  be  an  error  even  though  the  signs  were  changed,  is 
reproduced  in  section  c  of  the  diagram.  The  numerators  are  added 
and  the  denominators  multiplied.  A  fourth  error,  which  is  difficult 
to  explain,  appears  in  section  d,  the  numerators  being  subtracted 
and  the  denominators  added.    It  indicates,  however,  a  complete 


68 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


dependence  upon  method  or  formulae,  divorced  from  an  under- 
standing of  the  operation  to  be  performed. 


FRACTIONS  OP  tJNLIKE  DENOMINATORS 


In  Table  XXX  the  accuracy,  or  rather  the  degree  of  error,  with 
which  each  of  the  examples  in  Set  O  is  worked  by  the  eighth-grade 
pupils  in  Cleveland  and  Grand  Rapids  is  indicated.     The  facts  in 


TABLE  XXX 


Number  of  Pupils  Out  of  ioo  Failing  on  Each  Example  m  Set  0 
Eighth  Grades  of  Cleveland  and  Grand  Rapids  Compared 

Examples  in  Fractions 

City 

H+i 

i\+i 

3+A 

i\-l 

s-.. 

i-A 

ixg 

iXjg 

iXA 

{1-5-i 

i-i-lJ 

l?-i 

Total 

Cleveland 

Grand  Rapids. . 

40 
44 

40 
51 

49 
53 

51 

50 

48 
46 

40 
44 

13 
13 

II 
14 

10 
17 

49 
39 

37 
38 

44 
36 

432 
445 

Total 

84 

90 

102 

lOI 

94 

84 

26 

25 

27 

88 

75 

80 

877 

this  table  were  secured  as  follows:  From  the  records  of  the  eighth- 
grade  pupils  of  each  of  the  cities  there  were  taken  at  random  the 
records  of  100  pupils  who  had  attempted  all  twelve  examples.  It 
may  be  argued  that  such  a  method  necessarily  involves  a  selection 
of  either  a  superior  or  an  inferior  set  of  records.  That  such,  how- 
ever, is  not  the  case  to  any  marked  degree  is  shown  by  a  comparison 
of  the  percentage  of  accuracy  for  the  Cleveland  and  Grand  Rapids 
eighth  grades  as  represented  in  Diagram  11,  with  the  corresponding 
measures  of  accuracy  for  the  records  presented  in  Table  XXX. 
From  Diagram  11  we  find  the  percentages  of  accuracy  for  Cleveland 
and  Grand  Rapids  to  be  67 . 6  and  66 .  o,  respectively,  while,  as  com- 
puted from  the  table,  the  corresponding  percentages  are  64.0  and 
62.9.  It  is  probable,  however,  that,  even  though  a  superior  or 
inferior  group  were  selected,  the  distribution  of  errors  would  not 
be  far  from  valid. 

The  table  shows  great  variability  of  difficulty  among  the  12 
examples.  This  is  seen  at  once  to  be  due  to  the  composite  character 
of  the  set.  The  three  examples  representing  each  type  of  operation 
show  quite  close  agreement  in  the  number  of  errors  made  on  each. 


TYPES  OF  ERRORS 


69 


Likewise  the  differences  between  the  two  cities  are  not  great.  From 
this  table  the  average  percentage  of  error  has  been  determined  for 
each  of  the  four  types  of  fractions,  viz.,  addition,  subtraction,  mul- 
tiplication, and  division,  A  graphical  representation  of  these  facts 
appears  in  Diagram  20.  From  this  diagram  it  is  seen  that  sub- 
traction is  the  most  difl&cult  operation,  followed  in  order  by  addi- 
tion, division,  and  multiplication;  the  last-named  operation  is 
found  to  be  especially  easy,  while  there  are  no  great  differences 
among  the  other  three. 


"^fx  c/  fracft 


»/» 


/iJcfiirtOrr 

Xi /vis  ion 


a 

ta 

»o 

30 

40 

sc 

Diagram  20. — Showing  the  average  percentage  of  error  made  in  each  of  four 
types  of  fractions. 

In  the  discussion  of  the  types  of  errors  addition  will  first  receive 
attention.     In  Table  XXXI  there  is  shown  the  distribution  of  100 


TABLE  XXXI 

Frequency  of  Types  of  Error  in  Addition  of  Fractions  of  Unlike 
Denominators — Eighth  Grade 


Frequency 

TvPE  OF  Error 

Cleveland 

Grand 
Rapids 

Total 

Numerators  added,  denominators  added 

30 
23 

35 

2 

4 

2 

5 

46 

22 

10 

10 

8 

2 

2 

76 
44 
45 
12 

Numerators  multiplied,  denominators  multiplied 

Mistakes  in  " fundamentals" 

Confusion  of  sjnnbols 

Numerators  multiplied,  denominators  added 

Numerators  added,  denominators  multiplied 

Method  obscure 

12 

4 
7 

Total 

100 

100 

200 

70  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

errors  for  both  Cleveland  and  Grand  Rapids  eighth-grade  pupils. 
In  Diagram  21  there  are  five  typical  errors  as  the  work  was  actually 
done  by  the  pupils. 

II  J^_i2__4  li  i_L_lE  "_ui— II  Il4__L-i?  II4.—  II 
15     6     21     7         15     6     90         15     6     21         IS     6     90         15     6     66 

a  b  c  d  e 

DiAGSAM  21. — ^Typical  errors  made  in  adding  fractions  of  unlike  denominators 

The  most  frequent  error  is  the  one  found  to  occur  most  often  in 
the  addition  of  fractions  of  like  denominators,  viz.,  the  addition  of 
both  numerators  and  denominators  (see  section  a  of  the  diagram). 
Each  of  the  other  five  types  of  error,  with  the  exception  of  "mis- 
takes in  fundamentals"  and  "confusion  of  symbols,"  which  are 
easily  understood,  is  reproduced  in  sections  h,  c,  and  d  of  the  dia- 
gram. These  three  errors  are  also  the  typical  errors  found  in  the 
addition  of  fractions  in  Set  H.  There  were  several  errors  found 
which  it  was  impossible  to  analyze.  These  were  all  grouped  under 
"method  obscure."  An  error  which  was  very  rare,  but  interesting, 
appears  in  section  e  of  the  diagram.  This  pupil  inverted  the  divi- 
dend and  multiplied.  It  was  thus  an  incorrect  method  for  division, 
although  that  was  undoubtedly  the  method  influencing  the  pupil 
to  make  a  response  of  that  sort. 

A  comparison  of  the  records  made  by  the  Cleveland  pupils  and 
the  Grand  Rapids  pupils  shows  some  rather  clear  differences.  The 
Cleveland  pupils  seem  to  be  especially  weak  on  the  "fundamentals" 
when  they  are  used  in  connection  with  the  solving  of  fractions. 
That  is,  they  have  the  proper  method  of  dealing  with  the  fractions, 
but  make  mistakes  in  the  simple  combinations.  The  Grand  Rapids 
children,  on  the  other  hand,  have  a  relatively  stronger  predilection 
to  add  both  numerators  and  denominators  than  do  the  Cleveland 
children.     In  other  respects  the  two  groups  are  not  greatly  different. 

Turning  now  to  Table  XXXII  and  Diagram  22,  we  find  the 
facts  presented  for  the  subtraction  of  fractions  of  unlike  denomina- 
tors. The  table  shows  the  most  frequent  error  to  be  the  subtrac- 
tion of  both  numerators  and  both  denominators.  It  will  be  noticed 
that  this  error  has  about  the  same  frequency  as  the  corresponding 
error  in  addition,  viz.,  the  addition  of  both  numerators  and  of  both 


TYPES  OF  ERRORS 


n 


denominators.  This  error  may  be  of  two  kinds,  the  most  frequent 
being  represented  in  section  a  of  Diagram  22.  In  the  other,  the 
subtraction  is  in  the  same  direction  for  both  numerator  and  denomi- 
nator. Thus  5  —  2  would  give  3  for  the  numerator  and  6—21  would 
give  o  for  the  denominator,  instead  of  15,  obtained  by  subtracting 
the  other  way,  21  — 6,  as  in  section  a. 

TABLE  XXXII 

Fkequency  of  Types   of   Error   in   Subtraction   of  Fractions  of  Unlike 
Denominators — Eighth  Grade 


Frequency 

Type  of  Error 

Cleveland 

Grand 
Rapids 

Total 

Numerators  subtracted,  denominators  subtracted.  .  . 
Confusion  of  symbols 

20 

37 

24 

4 

4 

2 

3 
6 

59 
16 

5 
4 
2 
2 

7 
5 

79 

S3 

29 

8 

6 

Mistakes  in  fundamentals 

Numerators  added,  denominators  multiplied 

Numerators  added,  denominators  addet 

Numerators  subtracted,  denominators  multiplied. . . . 
Miscellaneous 

4 
10 

Method  obscure 

II 

Total 

100 

100 

200 

The  confusion  of  symbols  is  more  frequent  here  than  in  addition. 
This  difference  will  be  explained  in  comparing  Cleveland  and  Grand 
Rapids  results.  The  other  types  of  errors,  "numerators  added  and 
denominators  multiplied,"  "numerators  added  and  denominators 

6     21    15     5         6     21    126        6     21    27        6     21     126        6     21    105 

a  b  c  d  e 

Diagram  22. — ^Typical  errors  made  in  subtracting  fractions  of  unlike  denominators 


added,"  and  "numerators  subtracted  and  denominators  multiplied," 
are  shown  in  sections  b,  c,  and  d  of  the  diagram  and  therefore  need 
not  be  discussed.  To  include  certain  other  errors,  very  rarely 
made,  the  category  "miscellaneous"  has  been  introduced  into  the 
table.  In  the  last  section  of  the  diagram,  section  e,  there  is  repro- 
duced an  error  made  by  the  same  pupil  who  made  the  error  shown 
in  the  same  section  in  Diagram  21.    The  error  is  also  the  same,  the 


72  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

inversion  of  one  of  the  fractions,  followed  by  multiplication.  This 
incorrect  method  of  dividing  seems  to  be  quite  strongly  fixed  in  this 
individual. 

In  comparing  the  records  made  by  the  Cleveland  and  Grand 
Rapids  children,  the  dififerences  seem  to  be  greater  than  the  simi- 
larities in  the  three  types  of  errors  occurring  most  frequently.  The 
difference  between  the  frequency  of  "confusion  of  symbols"  in  the 
two  cases  is  explained  by  a  difference  in  the  organization  of  Set  O 
as  the  test  was  given  to  the  two  cities.  When  given  to  Cleveland 
the  set  was  composed  of  four  columns  of  examples,  3  to  each  column, 
and  the  3  examples  in  each  column  were  of  the  same  type.  The 
examples  in  the  first  column  were  addition,  those  in  the  second  sub- 
traction, those  in  the  third  multiplication,  and  those  in  the  fourth 
division.  Because  of  this  arrangement  the  principle  of  suggestion 
operated  in  this  set  as  it  did  in  Set  H,  previously  mentioned,  and 
caused  a  greater  frequency  of  confusion  of  symbols  for  Cleveland 
than  for  Grand  Rapids.  The  greater  frequency  of  mistakes  in 
fundamentals,  the  simple  combinations,  among  the  Cleveland  chil- 
dren has  been  commented  on  in  connection  with  the  examples  in 
addition.  The  Cleveland  children  likewise  seem  to  be  less  inclined 
to  subtract  than  to  add  both  numerators  and  both  denominators. 

Table  XXXIII  and  Diagram  23,  corresponding  to  the  tables  and 
diagrams  for  addition  and  subtraction,  indicate  the  character  of  the 
errors  made  by  the  eighth-grade  pupils  in  the  multiplication  of 
fractions.  The  most  frequent  error  in  performing  this  operation  is 
shown  in  section  a  of  the  diagram.  In  this  case  the  pupil  first  finds 
the  least  common  denominator  and  then  adds  the  resulting  numera- 
tors. This  is  really  the  method  used  for  adding  fractions,  and  a 
large  portion  of  these  errors  can  in  all  probability  be  accounted  for 
by  confusion  of  symbols.  The  next  most  frequent  error,  however 
(section  h,  Diagram  23),  represents  a  confusion  of  the  method  of 
addition  and  that  of  multiplication.  The  pupil  begins  with  the 
former  method  and  ends  with  the  latter.  Thus  he  finds  the  least 
common  denominator  and  then  multiplies  the  resulting  numerators, 
leaving  the  least  common  denominator  as  it  is  and  setting  it  down 
as  the  denominator  of  the  fractional  product.  The  other  mistake 
which  is  made  in  connection  with  the  least  common  denominator 


TYPES  OF  ERRORS 


73 


idea  (section  g,  Diagram  23),  though  much  less  frequently  encoun- 
tered, may  be  profitably  discussed  here.  The  least  common 
denominator  is  found,  and  then  the  resulting  numerators  are 
multiplied  and  the  least  common  denominator  squared  to  give  the 
numerator  and  denominator  of  the  product.  It  will  be  noted 
that  the  result  secured  in  this  way  is  not  incorrect,  but  the  method 
is  bad.  From  these  three  types  of  errors  it  is  evident  that  the  idea 
of  the  least  common  denominator  interferes  with,  and  highly  com- 
plicates, the  very  simple  method  of  multiplying  fractions. 


TABLE  XXXIII 

Frequency  of  Types  of  Error  in  Multiplication  of  Fractions  of  Unlike 
Denominators — Eighth  Grade 


Type  of  Erkor 


Freqxjency 


Cleveland 


Grand 
Rapids 


Total 


L.  C.  D.,  numerators  added 

L.  C.  D.,  numerators  multiplied 

Mistakes  in  fundamentals 

Numerators  added,  denominators  added 

Nximerators  added,  denominators  multiplied 

Inversion,  numerators  multiplied,  denominators  mul 
tiplied 

Inversion  of  results 

L.  C.  D.,  numerators  multiplied,  denominators  multi- 
plied  

Miscellaneous 

Method  obscure 


36 
28 
12 


25 

15 
12 

4 
8 

13 
10 


2 

I 
10 


61 
43 
24 
12 
II 

13 
10 

2 

9 

IS 


Total. 


The  remaining  types  of  errors,  represented  by  sections  c,  d,  e, 
and  /  of  the  diagram,  require  some  comment  at  this  point.  In  the 
iaist  of  these,  c,  we  have  the  addition  of  both  numerators  and  both 
denominators  again,  in  d  the  addition  of  the  numerators  and  the 
multiplication  of  denominators,  in  e  the  application  of  the  method 
of  division,  and  in  /  the  inversion  of  results.  The  last-named  error 
is  one  of  that  peculiar  type  already  represented  by  difficulty  in  the 
conception  of  zero  and  of  unity  noted  in  connection  with  the  simple 
multiplication  and  division  combinations.  The  table  shows  this 
error  to  be  comparatively  frequent  in  Grand  Rapids,  while  it  is 
not  represented  by  a  single  instance  in  the  records  studied  for 


74  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

Cleveland.    This  same  mistake  occurs  in  the  first  part  of  section  h 
of  the  diagram.    It  is  difl&cult  for  a  pupil  to  get  the  distinction 

20  I 

between  —  and  — ,  or  it  may  rather  be  that  the  pupil  tends  to 
I  20 

discard  the  i  when  it  appears  as  the  numerator  just  as  it  is 

commonly  discarded  when  it  appears  as  the  denominator.    This 

represents  a  very  interesting  type  of  confusion. 


0  10  30 

X 

6 

X^= 
zo 

=45      i_ 
30        6 

x^= 
10 

,4.        ±x^=^ 
16        6  10  60 

b 

c 

i 

6^10  18 

-r-X — ="20 
0   10 

6  10  900 

e 

f 

^4 
;sr^io  10 

i 

h 

Diagram  23. — ^Typical  errors  made  in  multiplying  fractions  of  unlike  denominators 

In  this  same  section  iji)  we  find  another  peculiar  type  of  error. 
The  work  on  all  these  examples  was  done  by  the  same  pupil.  The 
pupU  has  learned  to  cancel,  but  not  to  use  the  results  of  cancella- 
tion, except  in  the  first  of  the  examples,  where  it  seems  that  he  has 
used  the  "2''  because  of  inability  to  find  anything  else  to  put  down 
as  a  result. 

In  comparing  Cleveland  and  Grand  Rapids  certain  differences 
are  noted.  The  children  of  the  former  seem  to  be  more  inclined 
to  find  a  least  common  denominator  than  do  the  children  of  the 
latter  city.  This  is  partially  accounted  for  by  the  difference  in  the 
organization  of  the  test  when  given  to  the  two  groups,  already 
referred  to.  In  the  inversions  of  terms  and  of  results  Grand  Rapids 
monopolizes  all  the  errors  found.  In  other  respects  the  records 
from  the  two  cities  are  not  greatly  different. 


TYPES  OF  ERRORS 


75 


Turning  now  to  Table  XXXIV  and  Diagram  24,  we  find  the 
facts  presented  for  the  last  t5^e  of  examples  found  in  Set  0,  divi- 
sion. Failure  to  invert  the  divisor  (section  a,  Diagram  24)  seems 
to  be  the  most  frequent  error.  Another  interesting  and  logical 
error  is  the  dividing  of  the  numerator  of  one  of  the  fractions  by  the 
numerator  of  the  other  and  the  denominator  of  one  by  the  denomi- 
nator of  the  other  (section  b,  Diagram  24).  The  rule  is  that  the 
larger  quantity  is  divided  by  the  smaller  rather  than  the  term  of 
the  dividend  by  the  term  of  the  divisor.     A  very  interesting  example 


TABLE  XXXIV 

FREQtTENCY  OF  TYPES  OF  ErROR  IN  DIVISION  OF  FRACTIONS  OF  UnLIKE 

Denominators — Eighth  Grade 


Type  of  Erkok 


Frequency 


Cleveland 


Grand 
Rapids 


Total 


Failure  to  invert 

Mistakes  in  fundamentals 

Numerators  divided,  denominators  divided 

L.  C.  D.,  numerators  added 

Numerators  added,  denominators  added.  .  , 

Inversion  of  dividend 

L.  C.  D.,  numerators  subtracted 

Miscellaneous 

Method  obscure 

Total 


33 

27 

2 

14 
3 
2 

3 

2 

14 


26 

20 

20 

6 

4 

S 

3 

4 

12 


59 

47 

22 

20 

7 

7 

6 

6 

26 


of  this  error  is  found  in  section  g  of  the  diagram.  Here  the  pupU 
has  introduced  decimals  into  the  operation.  Another  error  deserv- 
ing mention  is  that  reproduced  in  section  e.  The  dividend  is 
inverted  instead  of  the  divisor.  The  other  types  of  errors  have 
all  appeared  in  connection  with  the  other  types  of  examples  and 
have  been  discussed;  hence  nothing  further  need  be  said  regard- 
ing them. 

There  also  seem  to  be  some  differences  between  Cleveland  and 
Grand  Rapids  in  the  division  of  fractions.  Children  of  the  former 
city  fail  to  invert;  that  is,  they  employ  the  method  of  multiplica- 
tion more  frequently  than  do  children  of  the  latter  city.  This  is 
probably  due  in  a  measure  to  the  difference  in  the  organization  of 


76  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

the  test  as  given  in  the  two  cities.  Cleveland  children  show  greater 
weakness  in  fundamentals  here  as  in  the  other  operations.  The 
addition  suggestion  already  discussed  is  found  to  operate  on  division 
for  the  Cleveland  children.  The  Grand  Rapids  children,  on  the 
other  hand,  seem  prone  to  divide  the  one  numerator  by  the  other 
and  the  one  denominator  by  the  other.  These  statements  cover 
the  chief  differences. 


20  . 

I 

20 

r 

21 

6 

a 

126 

20  . 

I 

21 

21  ' 

6 

d 

"27 

20 
21 

I 

'  6 

b 

20 
^3^ 

20 

:. 

21 

I 
6" 

e 

21 
120 

II  , 
12 

5_ 
8 

2.6 
1-4 

h 

ii_ 

2.1 
2.3 

21 

I 
6" 

20 
3-3 

20 
1. 

I 

=47 

21  ' 

6 

42 

c 

20  . 

I 

=  33 

21 

42 

Diagram  24. — ^Typical  errors  made  in  dividing  fractions  of  unlike  denominators 

SUMMARY 

1.  In  the  addition  of  the  simple  combinations  the  general  propo- 
sition seems  to  be  established  that  on  the  average  those  combina- 
tions whose  sums  exceed  ten  are  more  difficult  than  those  whose 
sums  are  less  than  ten.  To  ,this  general  statement  there  are  indi- 
vidual exceptions  which  indicate  the  formation  of  peculiarly  strong 
associations,  some  being  right  and  others  wrong.  These  peculiar 
associations  vary  among  different  groups.  This  would  indicate 
that  the  formation  of  the  association  is  to  be  accounted  for  in  terms 
of  the  experience  of  the  group  rather  than  in  the  character  of  the 
combination  itself. 

2.  In  the  simple  subtraction  combinations  "bridging  the  tens" 
is  found  to  be  a  relatively  much  more  difficult  operation  than  in  the 
addition  combinations.  Freakish  errors,  on  the  other  hand,  are 
found  to  be  less  frequent  in  the  former  than  in  the  latter.    The 


TYPES  OF  ERRORS  77 

understanding  of  the  meaning  of  zero  seems  to  accompany  the 
maturing  of  the  pupil.  This  is  indicated  by  a  relatively  large  per- 
centage of  errors  made  on  the  combination  i— o  by  fifth-grade 
pupils,  whereas  this  combination  presented  but  little  difficulty  to 
pupils  in  the  eighth  grade. 

3.  Practically  all  the  errors  made  in  the  simple  multiplication 
combinations  are  made  in  those  combinations  in  which  zero  enters 
as  one  of  the  terms.  Furthermore,  it  is  a  more  difficult  mental 
operation  to  multiply  a  quantity  by  zero  than  to  perform  the  reverse 
operation,  to  multiply  zero  by  the  quantity.  And  a  pupil  may  have 
difficulty  with  the  zero  in  the  simple  combinations,  yet  be  quite  able 
to  handle  it  in  the  more  complex  examples,  and  vice  versa.  In  the 
complex  multiplication  examples  the  most  frequent  error  is  made 
in  multiplying. 

4.  In  the  simple  division  combinations  the  most  frequent  error 
is  made  in  dividing  a  quantity  by  itself.  The  result  given  is  zero, 
showing  a  confusion  between  the  division  and  subtraction  processes. 
In  long  division  the  demand  for  multipUcation  accounts  for  most 
of  the  errors. 

5.  The  typical  errors  made  in  working  fractions  indicate,  as  a 
general  rule,  a  slavish  adherence  to  the  mechanics  of  fractions  and 
show  emphasis  upon  method  rather  than  upon  an  understanding  of 
the  process.  There  consequently  follows  a  great  deal  of  confusion 
of  methods  on  the  part  of  the  pupil. 

6.  In  the  addition  and  subtraction  of  fractions  of  like  denomi- 
nator there  is  a  tendency  to  add  both  numerators  and  denominators 
in  the  one  case  and  subtract  them  in  the  other. 

7.  In  the  working  of  fractions  of  unlike  denominator  those 
involving  subtraction  are  found  to  be  the  most  difficult,  followed 
in  order  of  decreasing  difficulty  by  those  involving  addition,  divi- 
sion, and  multiplication.  Multiplication  of  such  fractions  is  shown 
to  be  especially  easy. 

8.  In  the  application  of  each  of  the  fundamental  operations  to 
fractions  there  seem  to  be  certain  types  of  errors  which  recur  again 
and  again.  Careful  attention  on  the  part  of  the  teacher  to  these 
typical  errors  would  be  worth  while. 


CHAPTER  V 

A  COMPARISON  OF  THE  ARITHMETICAL  ABILITIES  OF  CERTAIN 
AGE  AND  PROMOTION  GROUPS 

One  of  the  great  values  of  a  standard  test  is  to  throw  light  on 
our  methods  of  instruction,  the  general  organization  of  our  courses 
of  study,  our  system  of  promotion,  and  so  on,  through  an  analysis 
of  what  children  do  under  these  different  influences.  The  present 
study  represents  an  attempt  to  compare  the  arithmetical  abilities 
or  attainments  of  certain  age  and  promotion  groups.  It  therefore 
falls  into  two  divisions,  closely  related  and  dealing  largely  with  the 
same  problem,  the  one  concerning  itself  with  differences  in  groups  of 
children  classified  according  to  age,  and  the  other  with  differences 
in  groups  classified  according  to  rates  and  causes  of  promotion  or 
non-promotion.  Although  these  two  divisions  of  the  study  are 
very  closely  related,  they  will  be  treated  separately  for  the  sake  of 
convenience. 

AGE   GROUPS 

The  purpose  of  this  division  of  the  investigation  is  to  find  out 
whether  or  not  there  are  any  differences  in  the  arithmetical  abilities 
which  accompany  differences  in  the  age  of  pupils  in  the  same  grade, 
that  is,  whether  the  under-age  group  is  at  all  different  from  the 
over-age  group,  or  whether  the  intermediate  or  normal  group  differs 
from  either  of  these.  Of  course  the  test  employed  is  quite  inade- 
quate to  indicate  all  differences  in  arithmetical  abilities,  but  in  so 
far  as  the  test  is  adequate  the  nature  of  the  differences,  if  any  exist, 
will  be  analyzed. 

METHOD 

The  data  upon  which  this  part  of  the  study  is  based  were 
secured  from  the  results  of  the  arithmetic  test  given  to  the  children 
in  the  B  sections  of  Grades  3-8  inclusive  of  the  Cleveland  schools. 
The  giving  of  the  test  has  already  been  discussed  and  therefore  need 
not  be  taken  up  here. 

78 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    79 

On  the  first  page  of  each  folder  the  age  of  the  pupil  taking  the 
test  was  called  for,  as  indicated  in  the  reproduction  of  the  test  in 
chapter  ii,  except  that  it  called  for  age  in  years  only,  not  in  years 
and  months  as  in  the  revised  test.  Thus  we  had  recorded  the  age 
of  each  of  the  children  taking  the  test.  It  was  therefore  possible 
to  group  the  pupils  in  each  grade  according  to  age. 

Now  it  will  be  remembered  that,  while  the  median  results  for 
Cleveland  as  a  whole  seemed  to  be  devoid  of  any  considerable 
inaccuracy,  there  was  some  doubt  as  to  the  accuracy  of  any  par- 
ticular record,  owing  to  the  fact  that  the  test  is  a  complicated  one 
involving  time  allowances  difficult  to  administer  exactly,  and  to 
the  fact  that  it  was  given  by  teachers  with  little  or  no  training  in 
giving  tests  of  this  sort.  Thus  it  is  evident  that,  if  our  comparisons 
are  to  be  valid,  some  method  must  be  adopted  which  will  eliminate 
those  errors  which  may  have  been  made  in  timing. 

Furthermore,  an  examination  of  the  results  of  any  standard  test 
secured  for  the  various  schools  and  classes  of  a  large  city  system 
shows  that  there  are  large  differences  from  school  to  school  and 
from  class  to  class  that  are  to  be  accounted  for  by  differences  in  the 
training  which  the  pupils  in  the  different  schools  and  classes  have 
received.  This  must  also  be  taken  care  of  by  our  method;  other- 
wise differences  between  two  age  groups  might  be  due  to  differences 
in  training  rather  than  to  differences  in  age. 

Thus  it  is  seen  that  there  are  two  factors  which  might  account 
for  differences  between  two  groups  that  must  be  eliminated.  The 
first  of  these  is  differences  in  giving  the  test.  Overtiming  or  under- 
timing  would  favor  or  prejudice  one  group  with  reference  to  another. 
The  second  of  these  is  differences  in  training.  One  group  may 
show  superiority  over  another  because  its  members  have  had  a  more 
effective  course  of  training.  In  order,  therefore,  that  any  compari- 
sons which  are  made  may  be  valid,  it  is  necessary  that  the  groups 
compared  be  homogeneous  as  to  the  conditions  under  which  the  test 
was  given  and  as  to  training.  Of  course  there  are  other  minor 
factors  which  may  have  influence,  but  it  is  believed  by  the  writer 
that  if  these  two  are  taken  care  of  the  comparisons  will  be  valid. 

After  an  examination  of  the  records  had  showed  that  it  would 
be  possible  to  secure  data  on  four  age  groups,  the  records  made  by 


8o  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

the  pupils*  of  a  grade  in  a  particular  school  were  divided  into  two 
groups  on  the  basis  of  sex;  then  each  of  these  was  thrown  into  four 
groups  on  the  basis  of  age.  Since,  however,  the  age  of  each  pupil 
was  given  in  years  only,  it  was  impossible  to  divide  the  boys  and 
girls  of  an  entire  class  into  four  equal  groups.  For  instance,  sup- 
pose we  have  a  third-grade  class  of  20  children.  It  is  probable 
that  10  of  these  will  be  boys  and  10  girls.  Of  the  10  boys  it  is  prob- 
able that  I  will  be  seven  years  old,  4  eight  years  old,  4  nine  years 
old,  and  i  ten  years  old  or  more.  Now  in  order  that  each  of  our 
four  age  groups  may  be  equally  influenced  by  the  giving  of  the  test 
to  this  class  and  by  the  training  which  the  class  has  received,  it  is 
necessary  that  this  class  be  equally  represented  in  each  group. 
Since  there  is  but  one  pupil  in  the  lower  age  group  and  but  one  in 
the  upper,  one  must  be  taken  at  random  from  each  of  the  inter- 
mediate groups.  The  same  method  is  followed  for  the  girls.  Thus 
from  this  class  of  20  pupils  but  4  boys  and  4  girls  have  been  taken, 
because  the  ages  were  given  in  years. 

This  method  of  selecting  pupils  for  each  of  the  age  groups  was 
continued  until  there  were  secured  records  from  50  boys  and  50 
girls  for  each  of  the  age  groups  in  each  grade  from  the  third  to  the 
eighth  inclusive.  This  made  a  total  of  100  records  for  each  group 
in  each  grade,  or  400  records  for  each  grade,  making  a  grand  total 
of  2,400  records,  upon  which  this  study  is  based.  In  order  to 
secure  this  number,  the  records  made  by  40-50  schools  were  ana- 
lyzed for  each  grade.  And  it  should  be  reasserted  that  the  four  age 
groups  in  each  of  the  grades  (100  pupils  to  the  group)  represent 
experience  in  taking  the  test  as  nearly  identical  as  it  is  possible  to 
make  it,  and  also,  after  allowing  for  differences  due  to  transfer  from 
one  school  to  another,  identical  training  so  far  as  training  in  the 
school  is  concerned. 

The  facts  concerning  the  ages  of  these  groups  in  the  several 
grades  are  given  in  Table  XXXV.  Here  the  average  age  of  each 
group  is  given.  And  it  should  be  added  that  the  range  of  the  ages 
in  any  one  group  is  practically  confined  to  two  years  except  in  the 
case  of  Group  IV,  in  which  the  range  is  about  three  years.  This 
means  that  while  the  ages  of  the  pupils  of  one  group  are  not  identi- 
cal, because  of  differences  in  the  same  grade  from  school  to  school 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    8i 

in  this  respect,  the  groups  are  quite  homogeneous  as  to  age.  The 
table  shows  that  the  difference  between  the  average  age  of  each 
group  and  the  average  of  the  next  older  group  in  each  grade  is  at 
least  a  year  in  every  instance,  and  in  some  cases  it  is  considerably 
more.    Thus  the  groups  are  seen  to  represent  real  age  differences. 

TABLE  XXXV 

Average  Age  of  Each  op  Four  Age  Groups  in  Grades 

3-8.  50  Boys  and  50  Girls  per  Group  in 

Each  Grade 


Grade 

Group 

I 

II 

ni 

IV 

3 

4 

5 

6 

7 

8 

7-7 
8.7 
9-7 

II. 4 
12. 1 

8.8 
9.8 
10.7 
II. 6 
12. s 
13-1 

9.9 
10.9 
12.0 
12.9 

13-7 
143 

II. 6 
12.7 
13-6 
14.4 
ISO 
iSS 

After  these  records  had  been  secured  they  were  all  carefully 
regraded,  lest  any  error  due  to  the  scoring  of  the  pupils  should 
prejudice  the  results.  They  were  then  tabulated,  both  the  number 
of  examples  attempted  and  the  number  of  examples  worked  cor- 
rectly in  each  of  the  sets  of  the  test.  And,  finally,  average  "rights," 
average  "attempts,"  and  accuracy  were  determined  for  each  age 
group  in  each  grade  for  each  of  the  sets. 


RESULTS 

The  detailed  facts  concerning  the  number  of  examples  worked 
correctly  by  the  four  groups  in  the  six  grades  appear  in  Table 
XXXVI.  In  order  that  the  table  may  be  made  perfectly  clear  to 
the  reader  an  explanation  is  necessary.  The  Roman  numerals, 
I,  II,  III,  and  IV,  represent  the  four  age  groups  in  each  grade. 
Group  I  is  the  under-age  group,  Groups  II  and  III  are  the  interme- 
diate or  normal  groups,  and  Group  IV  is  the  over-age  group. 
Keeping  this  explanation  in  mind,  we  read  that  in  the  third  grade 
Group  I,  the  under-age  group,  worked  correctly  on  the  average  16. 6 
examples  in  Set  A,  10. 7  in  Set  B,  and  so  on.     Group  II,  the  younger 


82 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


of  the  two  intermediate  groups,  averaged  15.9  examples  correctly- 
worked  in  Set  A,  9.1  in  Set  B,  and  so  on.    In  this  same  way  the 


TABLE  XXXVI 

Average  "Rights"  in  Each  Set  for  Each  of  Four  Age  Groups  m 
Grades  3-8.    Data  from  2,400  Pupils 


Set 


Third  Grade 


n 


m 


IV 


Fourth  Grade 


I 


n 


III 


IV 


Fifth  Grade 


I 

n 

III 

23.6 

22.8 

22.7 

19. 1 

18. s 

17.9 

16.3 

14.9 

15-5 

18.2 

16.6 

16.0 

6.3 

6.1 

6.1 

7.6 

6.9 

6.7 

4.8 

4-9 

4.8 

45 

4-7 

3-7 

2-3 

1.9 

1.8 

3-7 

3-7 

3-7 

6.7 

6.1 

5-8 

2-3 

2.4 

2.0 

3-2 

30 

2.8 

I.I 

I.O 

0.8 

0.2 

0.1 

0.2 

IV 


A. 
B. 
C. 
D. 
E. 

F. 
G. 
H. 
I., 
J 


K. 
L  . 
M. 

N. 
O. 


16.6 

10.7 

71 
7.8 
4-2 

2.6 
1.8 
0.8 
o-S 
1-7 

0.2 


iS-9 
9.1 
6.0 
S-8 
4.2 

I 

1-4 
0.7 
0.4 
1.6 


16.S 
10.4 

6.3 
6.9 

4-7 

^■3 
1-4 
0.8 
0.4 
1-7 


16 
9-3 
7.6 
6-5 
4-7 

1.9 
1.6 
1.0 
0.4 
1.6 


0.9 


0.8 


0.9 


19.9 

ISO 
14.0 

139 
5-6 

4-5 
3-7 
2.8 
1.2 
31 

3-9 
1-7 
2.6 

05 


20.4 

13  I 
13-8 

13s 
5-7 

4-7 
3-6 
2.0 
0.8 
30 

3-8 
1-5 
2-3 
0.4 


19.7 
13s 

12.3 
5-0 

4.6 
3-4 
2.5 
1.0 
2.9 

3-4 
1.2 
2.2 
03 


21.2 

143 
13.0 

12-5 

6.3 

4.8 
3-4 
31 
0.9 

3-4 

3-6 
1-4 
2.5 
0-3 


Sixth  Grade 


Seventh  Grade 


Eighth  Grade 


A. 
B. 
C. 
D. 
E. 

F. 
G. 
H. 
I. 
J- 

K, 
L. 
M 

N. 
O. 


I 

II 

III 

25.0 

25.2 

25.2 

21.0 

20.2 

193 

16.3 

19.6 

6.3 

17.7 

19s 

6.6 

17.4 
18.8 

6.3 

7.8 

7.8 

6.8 

S-3 
6.6 

5-3 
6.5 

S-i 
5-8 

30 

31 

2-5 

4.2 

4.2 

40 

8.6 

8.4 

8.2 

2-4 

2.6 

2.2 

3-8 

3.6 

31 

1-3 

1.4 

1.0 

4.6 

4-3 

3-4 

IV 


2S-3 
20.0 
18. 1 
19.2 
7.0 

7S 
S-2 
6.S 
2-5 
4-7 

7.6 
2-4 
3-7 
1.0 

3-5 


I 

n 

HI 

26.2 

28.3 

293 

22.4 

21. 1 

23.8 

17-4 

18.3 

19.4 

21.6 

22.3 

21.8 

7.7 

7.2 

7.6 

9.0 

8..'> 

8.6 

5.8 

%•(> 

6.1 

7.8 

7-9 

8.5 

4-3 

3.0 

35 

S.o 

4-7 

4-7 

10.7 

10.2 

9  9 

3-1 

2.8 

31 

4.8 

41 

4-2 

1.9 

1.6 

1.6 

5-9 

4.8 

4-4 

IV 


28.7 
22.3 
19.7 
21.4 

7-4 

8.0 
5-6 
8.2 
3-2 
4.6 

9.6 

2-3 

3-8 
1.2 

3  9 


24.2 
17.7 
16.0 
15.6 
6.4 

6.3 
4-3 
3  9 
1.8 
3-6 

5  9 

1-7 

2-5 

0.8 
0.2 


I 

n 

m 

IV 

30.1 

273 

30.1 

27. 

27.0 

25-5 

26.2 

24- 

18. 5 

18.6 

19.9 

18. 

23 -3 

22-5 

23.1 

22. 

8.2 

78.5 

7.6 

7- 

10.7 

9-4 

9.6 

8. 

6.8 

6.2 

6.0 

■>• 

95 

8.7 

8.4 

8. 

50 

4.4 

41 

3- 

6.1 

S-4 

5-4 

5- 

13.2 

12. 1 

II. 6 

II. 

4.0 

3-3 

31 

3- 

5-4 

4.6 

4-5 

4- 

2.6 

2.3 

2.0 

I. 

7-5 

6.0 

5-5 

4- 

scores  for  the  other  two  groups  in  the  third  grade  may  be  read,  as 
well  as  the  scores  for  all  four  groups  in  each  of  the  remaining  grades. 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    83 

A  glance  at  the  table  is  sufficient  to  show  that  there  is  a  tendency 
for  the  average  score  to  diminish  in  passing  from  Group  I  to 
Group  II,  from  Group  II  to  Group  III,  and  from  Group  III  to 
Group  IV  in  each  of  the  sets  in  each  of  the  grades.  There  are 
exceptions,  of  course,  and  the  differences  encountered  in  passing 
from  a  younger  group  to  an  older  one  vary  in  degree  with  the  sets 
and  with  the  grades. 

Since  the  more  important  facts  presented  in  Table  XXXVI  are 
presented  graphically  in  the  diagrams  which  follow,  we  shall  now 
pass  to  them.  In  these  diagrams  comparisons  are  made  between 
two  groups  only — Group  I,  the  under-age  group,  and  Group  IV, 
the  over-age  group — because  these  two  groups  represent  the 
extremes.  In  the  four  sections  of  Diagram  25  the  comparisons  are 
made  between  the  two  groups  throughout  the  six  grades  in  the  four 
sets  in  addition.  A,  E,  J,  and  M.  A  few  very  interesting  differences 
are  to  be  found  in  comparing  the  two  curves.  In  the  simpler  sets, 
A  and  E,  the  over-age  group  seems  on  the  whole  to  be  superior  to 
the  under-age  group,  while  in  the  more  complex  sets,  J  and  M,  and 
especially  in  the  latter,  the  superiority  of  the  under-age  group  is 
quite  marked.  There  also  seems  to  be  a  difference  in  the  relations 
of  the  two  curves  in  the  lower  and  the  upper  grades.  In  the  former 
the  differences  between  the  attainments  of  the  groups  are  less  in 
evidence  than  in  the  latter.  That  is,  the  diagram  would  indicate 
that  the  differences  between  the  under-age  pupils  and  the  over-age 
pupils  become  more  noticeable  as  we  proceed  upward  through  the 
grades;  and  this  is  especially  true  of  the  records  made  in  the  more 
complex  examples. 

Passing  now  to  Diagram  26  we  come  to  a  similar  comparison  of 
records  made  in  subtraction.  The  records  made  by  the  two  groups 
in  the  two  sets  of  examples  in  subtraction  are  here  graphically  pre- 
sented. The  same  conclusions  may  be  drawn  from  this  comparison 
as  were  drawn  from  the  comparisons  of  the  records  of  the  two 
groups  in  the  four  sets  of  addition,  viz.,  that  the  differences  are  less 
marked  in  the  simpler  than  in  the  more  complex  set  and  that  the 
superiority  of  the  under-age  group  increases  with  the  progress 
through  the  grades. 


84 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


The  records  made  in  the  three  sets  in  multiplication,  C,  G,  and  L, 
by  the  two  groups  of  pupils  under  comparison  are  shown  in  Dia- 


l^ro*tp  X  (i/rc/er/f^e) 

Groufs  js  COIN'S  y-  //y<r) 
s 


-f        5       6 
Grade 


^       S       6 
Grade. 


Diagram  25. — A  comparison  of  records  made  by  two  age  groups   through 
Grades  3-8  in  four  sets  in  addition  (A,  E,  J,  M). 

gram  27.    The  differences  already  noted  are  borne  out  here,  so  that 
no  discussion  is  necessary.    We  therefore  pass  to  Diagram  28,  in 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    85 

which  are  graphically  represented  the  facts  for  the  four  sets  in 
division,  D,  I,  K,  and  N.    These  graphs  deserve  especial  attention 


28 
26 
24 

ZC 
«'^ 

a 
o 


iSrottfo  T  CUnJer-  A^e. )— — — 
Group  W  {.D>^e  r  -A^  &)—  -  - 


4  S        C 

G-ro-dcs 


/€ 

F 

y        ^ 

B 

•4* 

/      / 
1    • 

fct 

/  / 
1  / 

01 

1/ 

<h 

J 

«-* 

/ 

V 

/ 

/ 

"^  .. 

n 

3L 

1 

0 

4         S        6 

GrcLoe. 


Diagram  26. — A  comparison  of  records  made  by  two  age  groups  through 
Grades  3-8  in  two  sets  in  subtraction  (B,  F). 


n 
J!" 

h 


Crot<f>SZ  iO*».rAgat-  —  —  * 


BraJa 


Bra.Je. 


-3 — ? — r— r 


DiAGKAM  27. — A  comparison  of  records  made  by  two  age  groups  through 
Grades  3-8  in  three  sets  in  multiplication  (C,  G,  L). 

because  they  so  clearly  represent  the  tendencies  noted  in  the  other 
sets.  It  should  be  remembered  that  Set  D  is  the  set  of  simple 
division  combinations,  Set  I  the  set  in  short  division.  Set  K  the 


86  STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

very  simple  set  in  long  division  examples,  and  Set  N  the  difficult 
set  in  long  division.    It  should  be  borne  in  mind,  further,  that  the 


£  rou/s  T  {Urtc/er-  Age  )  ~"^~~^ 
CrouftJC  carer -/i ye )—'-"--^ 


2/ 
/8 


s 

X                             / 

f 

/ 

/                   X 

/           x 

l« 

/         X 

2!-^ 
fc 

/    / 

^ 

/      / 

c^ 

/    / 

^Z 

/   /• 

\. 

/  • 

Q) 

/  / 

X 

/  / 

^, 

// 

/ 

/  / 

X  ^ 

//' 

r^ 

• 

c 

3        '? 


A^ 


^       6 
Grade. 


"1         ?         J        6         7         S 
^  ra  c/e 

Diagram  28. — A  comparison   of  records  made  by  two  age  gtoups  through 
Grades  3-8  in  four  sets  in  division  (D,  I,  K,  N). 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    87 


returns  on  accuracy  in  chapter  iii  showed  Set  I  to  be  more  difficult 
than  Set  K,  owing  to  the  introduction  into  the  examples  of  the 
former  set  of  the  operation  of  "carrying."  In  order  of  complexity 
and  of  difficulty,  therefore,  the  four  sets  should  be  arranged  thus: 
D,  K,  I,  and  N.  If  we  proceed  in  this  order,  it  is  very  clearly 
shown  that  as  we  pass  from  a  less  complex  to  a  more  complex  type 
of  operation  the  superiority  of 


Croup T  (,Undtr-/\gai 

Group  JS"  COyar'M^A)  — — — 


Sef  Z7 


8 


the  under-age  group  becomes 
more  and  more  obvious.  And 
furthermore,  as  before  indi- 
cated, these  graphs  very  plainly 
show  that  this  superiority  of 
the  under-age  group  increases 
with  progress  through  the 
grades. 

Diagram  29  is  of  interest 
because  it  is  a  diagram  present- 
ing the  facts  for  Set  O,  the 
most  complex  of  the  sets  and 
the  set  worked  with  the  great- 
est percentage  of  error.  In  this 
diagram,  more  than  in  any  of 
the  others,  the  superiority  of  the 
under-age  group  is  indicated. 

A  diagram  of  a  type  quite 
different  from  the  preceding  is 

found  in  Diagram  30,  in  which  the  attempt  has  been  made  to  make  a 
cross-section  of  the  records  of  Groups  I  and  IV  in  the  eighth  grade. 
In  order  to  do  this,  it  was  necessary  to  resort  to  the  system  of  weights 
derived  in  chapter  iii.  Under  this  system  the  average  "rights"  for 
each  of  these  groups  in  each  of  the  sets  has  been  converted  into 
terms  of  the  standard  "unit."  On  the  basis  of  the  average  number 
of  "units"  thus  made  in  each  set,  a  curve  is  drawn  to  represent 
each  group.  The  diagram  is  understood  if  it  be  borne  in  mind  that 
the  distance  of  the  curve  above  the  horizontal  axis  is  proportionate 
to  the  number  of  "units"  made  in  the  particular  set  indicated  by 
the  group  which  the  curve  represents.    This  diagram  brings  out 


4-      S      6 
Grade. 


Diagram  29. — A  comparison  of 
records  made  by  two  age  groups  through 
Grades  3-8  in  fractions  (Set  O). 


88 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


more  clearly  than  any  of  the  others  thus  far  examined  the  increas- 
ing superiority  qf  the  under-age  group  as  we  proceed  from  the  less 
to  the  more  complex  types  of  operation.  This  is  indicated,  with 
exceptions  of  course,  by  the  increasing  divergence  of  the  two  curves 
as  we  pass  from  left  to  right  in  the  diagram. 


Croup  T  WnJe.r-fi^4  '         '  ' 
Croufilff  Wv^r-flj/ci) 


DiAGiiAM  30. — Average  "units"  made  in  each  set  by  two  age  groups  in  eighth 
grade. 


This  same  matter  is  approached  from  a  slightly  different  angle 
in  Table  XXXVII  and  in  Diagram  31.  By  use  of  the  system  of 
weights  just  referred  to,  the  average  "rights"  made  by  each  of  the 
groups  in  each  of  the  sets  throughout  the  six  grades  has  been  con- 
verted into  the  terms  of  the  "unit."  The  15  sets  were  then  classi- 
fied into  six  groups  on  the  basis  of  complexity.  Into  the  first 
group  were  put  Sets  A,  B,  C,  and  D,  for  the  examples  of  these  sets 
are  clearly  the  most  simple  examples  in  the  test.  Into  the  second 
group  were  put  Sets  E,  F,  and  G,  representing  the  examples  of  the 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    89 

next  degree  of  complexity.  Set  I  was  not  included  in  this  group 
because  the  returns  seemed  to  indicate  that  it  is  of  much  greater 
difl&culty  than  any  of  the  other  three  sets.  Set  H  was  put  in  a 
class  by  itself  because  of  the  peculiar  type  of  reactions  made  to  it 
by  the  pupils.    Sets  I,  J,  and  K  were  then  put  into  the  fourth 


TABLE  XXXVII 

Average  Number  of  "Units"  Made  m  Certain  Groups  of  Sets  by  Each  op 
Four  Age  Groups  in  Grades  3-8.    Data  from  2,400  Pupils 


Set 

Third  Grade 

Fourth  Grade 

Fifth  Grade 

I 

n 

m 

IV 

I 

n 

III 

IV 

I 

II 

III 

IV 

A,  B,  C,  D 

49 

30 

3 

II 

5 

42 

26 

2 

10 

4 

47 

29 

3 

10 

5 

45 
29 

3 
10 

6 

75 
48 

9 
32 
31 

72 
50 
7 
29 
27 

70 
46 
8 
29 
24 

72 
51 
10 

31 
26 

92 
65 
15 
49 
45 
II 

277 

87 
62 
16 
45 
44 
6 

260 

85 
61 
12 
44 
38 
II 

251 

87 
59 
13 
43 
34 

E,  F,  G 

H 

I,  J,  K 

L,  M,  N 

0 

Total 

98 

84 

94 

93 

195 

i8s 

177 

190 

247 

Sixth  Grade 

Seventh  Grade 

Eighth  Grade 

I 

n 

III 

IV 

I 

II 

III 

IV 

I 

II 

m 

IV 

A,  B,  C,  D 

97 
67 
22 
60 
52 
26 

99 
68 
22 
60 
S3 
24 

96 
64 
19 
55 
43 
19 

97 
68 
22 
56 
48 
19 

103 
78 
26 
77 
69 
33 

107 

73 
27 
70 
59 
27 

112 
76 
29 
69 
61 
24 

no 

73 
28 
66 

49 
22 

117 
88 
32 
94 
86 
42 

112 
80 

29 
84 
75 
33 

117 

79 
28 
80 

68 
31 

112 

E,  F,  G 

76 
29 

76 
67 
27 

H 

I,  J,  K 

L,  M,  N 

0 

Total 

324 

326 

296 

310 

386 

363 

371 

348 

459 

413 

403 

386 

group  because  they  were  considered  to  be  less  complex  than  the  last 
three  sets  in  the  fundamentals,  L,  M,  and  N,  which  were  put  into 
the  fifth  group.  Set  O  was,  like  Set  H,  kept  by  itself  because  it  is 
different  from  the  other  sets,  the  examples  being  more  complex  and 
having  been  worked  with  a  larger  percentage  of  error  than  those 
of  the  other  sets.  Although  there  may  be  serious  question  con- 
cerning some  of  these  groups,  the  writer  is  of  the  opinion  that 
the  four  groups  composed  of  Sets  A,  B,  C,  and  D,  Sets  E,  F,  and  G, 
Sets  L,  M,  and  N,  and  Set  O,  respectively,  do  represent  groups  of 
examples  of  increasing  complexity. 


Croup  T  (l/ir  {/e.  f-/i^&y  ——— 


4        S       6 
Era4e 

Diagram  31. — A  comparison  of  average  numbers  of 
groups  of  sets  by  two  age  groups  through  Grades  3-8. 


'units"  made  in  certain 


8 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    91 


Thinking  this  last  statement  to  be  a  safe  proposition,  Diagram  3 1 
has  been  constructed,  in  which  the  average  records  made  by  the 
under-age  and  the  over-age  groups  in  these  four  groups  of  sets  are 
compared.  Thus,  as  we  pass  from  one  section  of  the  diagram  to 
the  next,  we  proceed  from  records  made  in  sets  of  simple  examples 
to  records  made  in  more  complex  examples.  Here  again  the  fact 
is  very   clearly   brought 


t^rOiAf?  I  (.Uncfer- A^a) 


out  that  the  superiority 
of  the  xmder-age  group  is 
not  at  all  conspicuous  in 
the  simpler  operations, 
but  becomes  more  and 
more  marked  as  the  more 
complex  operations  are 
encountered. 

One  more  diagram 
should  be  presented 
before  we  leave  this  phase 
of  the  problem  for  the 
purpose  of  bringing  out 
more  clearly  the  dififer- 
ences  met  with  as  prog- 
ress is  made  through  the 
grades.  In  order  to  bring 
the  cumulative  force  of 
records  made  in  the 
entire  test  to  bear  on  this 
problem,  the  average 
numbers  of  "units"  made 

by  each  of  the  age-groups  in  the  15  sets  are  added  to  get  a 
single  score  to  represent  each  group  in  each  grade.  A  graphic 
representation  of  this  comparison  is  shown  in  Diagram  32,  from 
which  it  is  plain  that  the  difference  between  the  total  scores  made 
by  the  under-age  and  the  over-age  groups  becomes  consistently 
greater  (except  at  the  sixth  grade)  from  grade  to  grade. 

Before  closing  the  comparison  between  the  age  groups  on  the 
basis  of  the  number  of  examples  worked  correctly,  two  summary 


«« 

^zo 

390 

V) 

3i,0 

330 
300 
210 

V 

2W 

N^ 

210 

180 

ISO 
120 
90 

<>0 

JO 

^ 

0 

T 


■5 


4  S         6  '4 

Diagram  32. — A  comparison  of  the  average 
numbers  of  "units"  made  in  all  sets  through 
Grades  3-8  by  two  age  groups. 


92 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


tables  should  be  presented,  Tables  XXXVIII  and  XXXIX.  In 
the  first  is  presented  the  average  number  of  "rights"  made  in  each 
set  by  the  four  age  groups  in  Grades  3-8  combined.  By  applying 
our  own  system  of  weights  to  this  table  we  obtain  the  second,  in 

TABLE  XXXVin 

Average  "Rights"  in  Each  Set  for  Each  of  Four 
Age  Groups  in  Grades  3-8  Combined 


Set 

Average  Rights 

I 

n 

m 

IV 

A 

B 

C 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

23.6 
19.2 
14.9 
17-4 
6.4 

7.0 
4-7 
5-3 
2.7 
4.0 

7.2 

2-3 

3S 
1.2 
30 

233 
17.9 
14.9 
16.7 
6.2 

6.5 
45 
S-i 

2-3 

3-8 

6.8 
2.1 

31 
I.I 

2.5 

23-9 
18. 5 
^5-3 
16.5 
6.2 

6.4 
4.4 
S.o 
2.2 
3-7 

6.5 
1.9 

3.0 

I.O 

2.3 

23.9 
18.0 

15.5 

16.3 

6.5 

6.2 

4-3 
5.2 
2.1 
3-8 

6.3 
1.8 
3.0 
0.9 
2.1 

which  is  found  a  statement  of  the  average  number  of  "units"  made 
in  the  entire  test  of  15  sets  by  each  of  these  age  groups  in  the  com- 
bined six  grades.    These  tables  bring  out  nothing  that  has  not 

TABLE  XXXrX 

Average  "Units"  Made  in  Each  of  Four  Age  Groups 
IN  All  Sets  by  Grades  3-8 


Group 

I 

II 

III 

IV 

Average  "units"  made. 

290 

272 

265 

262 

already  been  discussed,  but  merely  present  in  summary  form  what 
has  already  been  included  in  the  other  tables.  They  may  therefore 
be  passed  by  without  further  comment. 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS 


93 


Now,  to  sum  up  briefly  the  facts  that  have  been  discovered  with 
reference  to  the  number  of  examples  worked  correctly  by  the  pupils 


TABLE  XL 

AvEEAGE  "Attempts  "  in  Each  Set  for  Each  of  Four  Age  Groups  in  Grades  3-8. 
Data  from  2,400  Pupils 


Set 


A. 
B. 
C. 
D. 

E. 

F. 

G. 

H. 

I. 

J. 


K. 
L.. 
M. 

N. 
O. 


A. 
B. 
C. 
D. 
E. 

F. 
G. 
H, 
I. 
J- 

K 
L. 
M 

N, 
O. 


Third  Grade 


17-3 

ll-S 

8.6 

9.0 

4.9 

4.0 
2.6 
1.8 

1-4 
2.9 


II        III        IV 


16.6 
9.9 
7.2 
71 
4-7 

3-5 
2.4 

1-5 
i-S 
2.8 


1.9 


17. 1 
II. I 

7-5 
8.2 

5-3 

3-7 
2.3 
1.8 

1-3 
3-2 


17. 1 

10.6 

9.0 

7-9 

5-4 

3-8 
2.8 
1.8 
1.6 
3-2 


2-3 


Sixth  Grade 


I 

II 

m 

25.2 

254 

25 -7 

21.3 

20.5 

19.7 

18.4 

20.0 

19.6 

20.0 

20.0 

19.4 

6.7 

7.0 

6.8 

8.5 

8.6 

8.0 

S-9 

6.2 

5-8 

9.2 

9-3 

8.9 

3-8 

3-8 

3-4 

S-5 

s-o 

5-3 

9-3 

9.0 

9.0 

30 

3-9 

3-7 

4-9 

5-2 

4.6 

1.9 

2.2 

2.0 

7.2 

75 

7-3 

IV 


25-7 
20.5 
20.7 
20.0 
7-5 

8.7 
6.2 
9.8 
3-6 
6.1 


Fourth  Grade 


I 

II 

III 

20.1 

20.6 

19.9 

154 

134 

14. 1 

15.0 

^53 

145 

U-S 

14 -5 

131 

6.1 

6.2 

S-S 

5-7 

6.0 

6.0 

4-3 

4-3 

4.2 

3-7 

30 

3-4 

2.2 

1.9 

2.1 

4-3 

45 

4-4 

4.8 

4.6 

4.4 

30 

2.9 

2.7 

3-7 

3-5 

3f 

1-3 

1.2 

1-3 

0-5 

OS 

03 

21.7 
ISO 
14-7 
13 -7 
71 

6.6 

4-4 
4.0 
2.2 
S-o 

4-9 
3-3 
4.2 
1.6 
0.2 


Seventh  Grade 


I 

II 

III 

26.4 

28.6 

29.7 

22.7 

21.4 

243 

19-3 

21.0 

22.5 

21.9 
8.2 

22.9 
7.8 

22.5 
8.1 

9.9 
6.6 

9.8 
6.S 

9.9 
6.9 

10. 1 

10.9 

II. 7 

4-9 
6.S 

4.3 
6.1 

4-4 
6.3 

II. 4 

II. 0 

10.7 

4-4 
6.1 

4.6 
5-5 

4.6 
S.8 

2.4 
8.4 

2.4 
7-9 

2-5 

8.4 

IV 


29.0 
22.9 

22.8 

22.2 

8.1 

95 

6.9 

II. 4 
4-4 
6.3 

10.8 
4-4 
S-7 
2-3 
8.0 


Fifth  Grade 


I 

II 

III 

24.0 

22.9 

22.9 

193 

19.0 

18.3 

17.6 

16.7 

17.2 

18.6 

17.2 

17.0 

6.8 

6.6 

6.7 

8.3 

7-7 

7-6 

S-S 

5-6 

5-6 

6-3 

6.S 

6.0 

31 

2.7 

2.8 

50 

4.8 

S-i 

7.2 

6.8 

6.7 

3-7 

3-6 

3-6 

45 

4.2 

4-3 

1.9 

1.6 

1-7 

I.O 

0.8 

1-3 

Eighth  Grade 


I 

II 

m 

304 
27.2 

27.7 
25.8 

304 
26.  s 

21.0 

21.2 

22.3 

23-8 
8.S 

23.1 
8.0 

239 

8.2 

II. 6 

10.5 

10.6 

7-S 

7.0 

6.9 

II. 9 

10.8 

II. 8 

5-6 

S-i 

S-o 

7-4 

7.0 

7-1 

I3-S 

12.6 

12.4 

5-3 
6.8 
30 

S-o 
6.1 
2.8 

4-7 
5-9 
2.6 

9-3 

8.9 

8.7 

24-5 
18.2 
18.0 
16.7 
6.9 


71 
3-4 
4.2 

1-7 
1-7 


IV 


28.1 
24.7 
21.4 
23.1 
8.0 

9.9 

6.8 

II. I 

4-5 
6.8 

II. 8 
4.8 
6.0 

2-5 

8.4 


in  these  four  age  groups,  it  may  be  said:   (i)  that  there  are  differ- 
ences, (2)  that  on  the  average  the  younger  groups  are  superior  to 


94 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


the  older,  (3)  that  this  superiority  is  more  marked  in  the  later 
than  in  the  earUer  grades,  and  (4)  that  it  is  also  more  marked 
in  the  handling  of  the  more  complex  than  in  the  handling  of  the 
simpler  types  of  examples. 

We  pass  now  to  an  examination  of  the  records  made  by  the 
pupils  in  these  four  age  groups  for  the  purpose  of  discovering  differ- 
ences in  the  number  of  examples  attempted  in  the  various  sets. 
These  facts  are  presented  in  detail  in  Table  XL.  Since  this  table 
is  identical  in  form  with  tables  already  explained,  our  attention  may 
be  directed  at  once  to  the  facts  themselves.  A  glance  is  sufficient 
to  show  that  no  such  differences  are  to  be  found  in  the  comparison  of 
the  "attempts"  made  by  the  four  groups  as  were  found  in  the  com- 
parison of  the  "rights."  Especially  is  it  true  that  in  the  simpler 
sets  there  is  no  tendency  for  the  under-age  pupils  to  attempt  more 
examples  than  do  the  over-age  pupils.  In  the  more  complex  sets, 
however,  there  is  such  a  tendency,  especially  in  the  upper  grades, 
but  to  a  much  less  marked  degree  than  in  the  case  of  the 
"rights." 

Let  us  therefore  turn  to  Table  XLI  and  Diagram  33,  in  which 
are  presented  the  total  scores  attempted  in  the  entire  test  by  each 
of  the  groups  in  each  of  the  grades.    This  total  score  has  been 

TABLE  XLI 

Average  "Units"  Attempted  in  All  Sets  by  Each  op 
Four  Age  Groups  in  Grades  3-8 


Giade 

Group 

I 

II 

III 

IV 

3 

4 

s 

6 

7 

8 

133 

250 
327 
390 
454 
517 

121 
243 
307 
403 
447 
486 

130 
239 
3" 
386 
466 
489 

136 
267 
312 
412 

454 
469 

determined  by  using  the  system  of  weights  already  employed  in 
obtaining  total  scores  of  "rights."  The  diagram  shows  very 
clearly  that  there  is  no  clear  difference  between  the  two  extreme 
groups.    In  three  of  the  six  grades  the  over-age  group  attempts  more 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    95 


"units"  than  does  the  under-age  group,  while  in  only  two  grades 
is  the  reverse  the  case. 

In  Tables  XLII  and  XLIII  the  facts  concerning  attempts  are 
put  in  summary  form.     In  the  first  table  we  have  the  average 
number  of  examples  attempted  in  each  set  by  each  of  the  four  age 
groups  in  Grades  3-8 
combined,    while   in   the  pr0upXClJn(/&r-ft3«:y    "   • 

second    table    these    set  $roupisiaire.r-/ig&y 

averages  are  reduced  to 
a  single  score  by  the  use 
of  the  system  of  weights. 
Table  XLII  shows  no 
clear  tendency  of  any 
one  group  to  take  the 
lead;  one  group  forges 
ahead  in  one  set  only  to 
fall  back  in  another. 
Table  XLIII  likewise 
shows  no  differences 
worth  considering. 

In  summary  it  may 
therefore  be  said  that  the 
study  reveals  no  clear 
differences  between  any 
two  of  the  four  age 
groups  in  numbers  of 
examples  attempted  in 
the  various  sets. 

A  third  phase  of  the 
problem   now  presents 

itself,  although  it  is  implied  in  what  has  gone  before,  and  that 
is  the  question  of  accuracy.  Of  course,  since  it  has  been  found 
that  the  under-age  pupils  excel  in  "rights"  and  on  the  average 
attempt  no  more  than  do  the  over-age  pupils,  it  necessarily  follows 
that  the  former  have  attained  a  greater  degree  of  accuracy.  How- 
ever, since  this  is  only  a  general  impression,  it  will  be  worth  while 
to  make  a  special  study  of  accuracy. 


SIO 

/ 

48C 

/ 

4SQ 

/ill^a^s 

A^ 

4Z0 

y  / 

^i  390 

/  / 

^   360 

^330 

/t 

41 300 

/j 

•%, 

«^   270 

// 

\  2^0 

1- 

iJi   180 

// 

// 

if 

\I50 

1/ 

?  uo 

V. 

^    90 

\ 

^    60 

30 

0 

6 


8 


Z  3  ^  S 

Diagram  33. — A  comparison  of  the  average 
numbers  of  "units"  attempted  in  all  sets  through 
Grades  3-8  by  two  age  groups. 


96 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


In  Table  XLIV  there  is  presented  the  percentage  of  accuracy 
made  in  each  set  by  each  of  the  four  age  groups  in  Grades  3-8.  An 
examination  of  this  table  shows  the  differences  to  be  of  about  the 


TABLE  XLII 

Average  "Attempts"  in  Each  Set  for  Each  of  Four 
Age  Groxtps  in  Grades  3-8  Combined 


Set 


A. 
B. 
C. 
D, 
E. 

F. 
G. 
H, 
I. 
J 

K 
L. 

M 
N, 
O. 


Average  Attempts 


n 

III 

IV 

•9 

23.6 

243 

24- 

.6 

18.3 

19.0 

18. 

.6 

16.8 

173 

17- 

.0 

17s 

174 

17- 

.8 

6.7 

6.8 

7- 

.0 

7-7 

7.6 

7- 

■4 

5-3 

5-3 

5- 

.2 

7.0 

7-3 

7- 

•5 

3-2 

3-2 

3- 

•3 

51 

52 

5- 

•7 

7-3 

7.2 

7- 

•4 

3-3 

3-2 

3- 

•7 

4-4 

4-4 

4- 

•7 

1-7 

1-7 

I. 

•4 

4-3 

4-3 

4- 

same  order  as  those  found  in  the  comparison  of  the  groups  in  the 
number  of  examples  worked  correctly.  In  the  simpler  sets  there 
is  not  a  great  deal  of  difference  between  the  under-age  and  the 

TABLE  XLIII 

Average  Units  Attempted  in  Each  of  Four  Age 
Groups  in  All  Sets  by  Grades  3-8 


Group 

I 

11 

III 

IV 

Average  "units"  attempted 

345 

335 

337 

342 

over-age  pupils,  while  in  the  more  complex  examples  the  difference 
becomes  accentuated.  The  differences  also  appear  to  be  greater  in 
the  upper  than  in  the  lower  grades,  although  in  this  respect  there 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS     97 

is  less  dijfference  between  the  earlier  and  the  later  grades  than  was 
found  to  be  true  in  the  case  of  the  "rights." 

TABLE  XLIV 

Percentage  or  Accuracy  in  Each  Set  for  Each  of  Four  Age  Groups  in 
Grades  3-8.    Data  from  2,400  Pupils 


Set 


Third  Grade 


II 


III 


IV 


Fourth  Grade 


II 


III 


IV 


Fifth  Grade 


II 


III 


rv 


A. 

B. 

c. 

D. 
E. 

F. 

G. 

H. 

I. 

J 


K. 
L.. 
M. 
N. 
O. 


A. 

B 

C. 

D. 

E. 

F. 

G. 

H. 

I. 

J 


K. 
L. 
M. 

N. 
O. 


96.2 

93  o 
82.0 
85.8 
86.2 

645 
67.6 
46.0 
31.2 
58.4 

28.6 


96.1 
92.2 

83-5 
82.3 

89-5 

SO. 9 
60.9 
45-4 
25.2 

56.5 
55-6 


96. 1 
94.0 
84.0 

84.4 
87.8 

62.2 
60.8 

43-5 
28.0 

54-5 


94.2 
88.4 
84.7 
82.2 

85.7 

49-3 
S7-3 
52.2 
27.7 
50.2 


44.1 


43-4 


46.7 


47-4 


98.9 
97-3 
93-2 
95-8 
92.4 

79-3 
85.0 
74.8 
54-8 
72.1 

82.0 
56.0 
70.4 
40.9 


97.6 
903 
93  o 
91. 1 

77.0 

839 
68.2 

41-5 
6S-9 

84.0 
52.4 
63 -9 
311 


98.7 
95-2 
91.4 
93-9 
90-3 

76.1 
79.8 
73-3 
48.5 
64.4 

76 

43-8 
59-6 
25.2 


97-9 
95  S 
88.1 
91.2 
89.9 

72.6 

77-7 
78.8 
41. 1 
67.1 

73-3 
42.0 
61. 1 
21.0 


98. 
99. 
93- 
97- 
92. 

90. 
86. 
71- 
74- 
73- 

93- 
63- 
71. 
60. 
16. 


•5 

99. 

.1 

97- 

.0 

89. 

•S 

96. 

•7 

91. 

.8 

89. 

•  7 

88. 

•  4 

73- 

•  4 

68. 

•S 

76. 

•3 

89. 

•5 

64. 

.6 

71- 

•S 

60. 

■5 

14- 

97-9 

90. S 

93-8 
91.0 

87.9 
85.8 
62.6 

633 
74.0 

86.2 
57-2 
65-9 
47-9 
II. 9 


Sixth  Grade 


Seventh  Grade 


Eighth  Grade 


99-3 

"   7 


99 


III 


IV 


98.3 

97.6 

87.1 

95 

92.9 

86 

84 
66 
68.1 
77-9 

83.6 

58.5 
67.7 

50.5 
44-3 


II 


99.1 
98.6 
87.0 

97-3 
930 

87-5 
8S-3 
72.4 

83 -4 
77.0 

92.6 
61. 5 
73  I 
66.4 
60.7 


III 


98.5 
97-9 
86.3 
96.8 
93-5 

87.1 
88.2 
72.8 
78.9 
751 

92.6 
66.7 
71.2 
634 
530 


IV 


99.0 

99-3 
88.1 
97.8 
95-9 

92.4 
89.8 

79-7 
87.8 
82.2 

97-3 
76.1 
80.1 
86.9 
81. 1 


III 


99.0 
98.8 
89.1 
96.6 
93  o 

90-5 
87.3 
71.0 
81.8 
76.0 

935 

64.7 
76.4 
77-3 
63 -7 


97.1 
88.7 
93-2 
92.9 

82.2 
80.1 

659 
65.0 

714 

83.0 
50- 7 
58.4 
46.5 
8.7 


IV 


In  Diagram  34  the  under-age  and  the  over-age  groups  are  com- 
pared through  the  six  grades  in  Sets  L,  M,  N,  and  O.    These  graphs 


98 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


100 

9S 
10 
25 
80 
7S 
70 
65 

io 

\^^ 

5 -CO 

30 
Z5 

15 
10 

5 

0 


Croup T  (Unc/er-A^e)    

CroupTS  iOrsr-  -/Ij^e) 

lOu 


Set  L 


y^ 


f5 

as 

80 
IS 
70 
65 
^0 

ss\ 

So 

40 
35 
30 
2S 
20 
15 
10 
5 
o 


Ssf  /Vf 


/oo 
95 
90 
B5 
80 
7s 

is 

«r 

;i 

Ki40 
u 
^3S 

30 

2S 

to 

s 

0 


S^ir  N 


4        S 
UraJa 


8 


100 

95 

fo 

S»i  u 

8S 

80 

y 

7S 

/ 

10 

/ 

65 

^/^ 

60 

f 

S5 
50 
45 
40 

1                     • 
/                   • 

1         ^^ 

/  / 

35 

/  / 

30 

// 

Z5 

// 

20 

// 

IS 

'/ 

to 

/ 

5 

X) 

5 
■exJe. 


6 


Diagram  34. — A  comparison  of  percentages  of  accuracy  made  in  Sets  L  (mul- 
tiplication), M  (addition),  N  (division),  and  O  (fractions)  by  two  age  groups  through 
Grades  3-8. 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    99 

show  very  distinct  differences  between  the  two  groups,  and  these 
differences  seem  to  be  most  marked  in  Sets  N  and  O. 

On  the  basis  of  the  average  number  of  "units"  made  in  the 
entire  test  by  the  four  groups  in  the  several  grades,  presented  in 
Table  XXXVII,  and  the  average  number  of  "units"  attempted, 
presented  in  Table  XLI,  it  was  possible  to  determine  the  percentage 
of  accuracy  made  by  each  of  the  age  groups  in  the  entire  test. 
These  measures  of  accuracy  are  presented  in  Table  XLV  and  in 


TABLE  XLV 

Average  Accuracy  in  All  Sets  by  Each  of  Four  Age 
Groups  in  Grades  3-8 


Group 

I 

n 

III 

IV 

3 

4 

s 

6 

7 

8 

73-7 
78.0 

84.7 
83.1 
85.0 
88.8 

69.4 
76.1 

84.7 
80.9 
81.2 
85.0 

72.3 

74- 1 
80.7 
76.7 
79-6 
82.4 

68.4 
71.2 
79.2 
75-2 
76.7 
82.3 

TABLE  XLVI 

Average  Accuracy  for  Each  Set  for  Each  of  Four 
Age  Groups  in  Grades  3-8  Combined 


Average  Accuracy 

I 

II 

in 

IV 

A 

B 

C 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

98.6 
98.1 
89.6 
96.6 
93-3 

87.5 
86.4 
74-3 
76.8 

7S-5 

93-3 
673 
74.0 
71. 1 
693 

98. 5 
97-7 
88.4 
95-6 
92.4 

84.7 
85.0 
72.8 
72.6 
73-2 

92.3 
62.4 
68.9 

65s 
59-2 

98.3 
97.4 

88.6 

94.9 
91.6 

84.2 

84.3 
68.3 
69.6 
71.7 

89.9 
59-6 
67.6 
57-2 
52.2 

98.1 
96.6 

87.4 
94.2 
91.2 

80.6 
80.3 
71. 1 

64.9 
71.2 

86.3 
S4.8 
64.1 
51.6 
47-3 

lOO        STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


Diagram  35.  In  a  previous  paragraph  the  comment  was  made  that 
differences  in  accuracy  between  the  two  extreme  age  groups  did  not 
increase  in  the  same  degree  with  progress  through  the  grades  as 
was  found  to  be  true  in  the  case  of  differences  in  "rights."  This 
point  is  brought  out  very  clearly  in  Diagram  35,  which  shows  that, 

__  while  there  is  some  slight 


All  Seis 


increase  in  the  superiority 
in  accuracy  of  the  under- 
age group  over  the  over- 
age group,  as  we  pass  up 
through  the  grades  the  in- 
crease is  of  small  signifi- 
cance. 

Table  XL VI  is  a  sum- 
mary in  which  there  is 
presented  the  average 
accuracy  made  in  each  set 
by  the  four  groups  in  the 
six  grades  combined.  An 
examination  of  this  table 
shows  it  to  be  quite  re- 
markable, for  there  are 
only  two  cases  in  the  entire 
table  where  passing  from 
the  percentage  of  accuracy 
made  in  a  particular  set 
by  one  of  the  age  groups 
to  the  percentage  of  accu- 
racy made  by  the  next 
older  group  is  not  accom- 
panied by  a  decrease  in 
accuracy.  The  two  exceptions  are:  (i)  in  Set  C  the  accuracy 
of  Group  II  is  88.4  per  cent  and  that  of  Group  III  is  88 . 6  per  cent; 
(2)  in  Set  H  the  accuracy  of  Group  III  is  68 . 3  per  cent  and  that  of 
Group  IV  is  71 . 1  per  cent.  From  this  it  is  certainly  evident  that 
on  the  average  the  younger  pupils  of  a  grade  are  more  accurate  than 
the  older  pupils. 


9S 
90 
SS 
80 
TS 
70 

«l 

60 
5S 

\^ 

Z5 
26 
IS 

'    10 

s 

0 


3  4  5  (>  7  5 

C  rmJe. 

Diagram  35. — A  comparison  of  percentages 
of  accuracy  made  in  all  sets  by  two  age  groups 
through  Grades  3-8. 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    loi 

The  facts  presented  in  Table  XL VI  for  Groups  I  and  IV  are 
presented  in  Diagram  36  in  graphic  form.  The  one  thing  empha- 
sized by  this  diagram,  in  addition  to  the  mere  fact  that  the  under- 
age group  is  more  accurate  than  the  over-age  group,  is  that  the 
difference  becomes  more  marked  as  we  pass  from  the  simpler  to 
the  more  complex  examples,  until  the  greatest  difference  is  found 
in  Set  O,  fractions. 


Group  T  (i/nJ^tr-Aj^ 
Groua  27"  W^sr-  M§e) 


A     ^      2     T      TT      7      //      j"     D      X     Z 

Sa-t 


/w 


A*    '  a 


DiAGiLAM  36. — Average  accuracy  made  in  each  set  by  each  of  two  age  groups 
in  Grades  3-8  combined. 


Table  XL VII  presents  a  final  statement  of  the  relative  accuracy 
of  the  four  groups  in  all  the  sets  in  Grades  3-8  combined.  These 
facts  merely  emphasize  what  has  already  been  said,  and  they  need 
be  discussed  no  further. 

As  a  brief  summary  of  the  facts  relating  to  the  accuracy  of  the 
four  age  groups  in  the  tests  it  may  be  said :  (i)  that  there  are  differ- 
ences; (2)  that  these  differences  show  the  younger  pupils  to  be  on 
the  average  more  accurate  in  their  work  than  the  older  pupils; 


I02        STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

(3)  that  these  differences  are  on  the  whole  quite  uniform  from  grade 
to  grade;  and  (4)  that  the  differences  are  more  evident  in  the  more 
complex  than  in  the  simpler  examples. 


TABLE  XLVII 

Average  Accuracy  in  Each  of  Four  Age  Groups  in 
All  Sets  in  Grades  3-8 


Group 

I 

n 

m 

IV 

AccuraQT 

84.0 

81.3 

78.8 

76.8 

PROMOTION   GROUPS 

The  purpose  of  this  second  division  of  the  investigation  is  to 
throw  some  light  on  promotion  practices,  as  found  in  Grand  Rapids, 
and  their  relations  to  arithmetical  attainments.  Although  the 
paucity  of  the  data  has  made  it  quite  impossible  to  make  the  case 
conclusive  in  any  instance,  the  attempt  is  made  in  this  study  to  do 
four  things:  (i)  to  determine  differences  in  arithmetical  attainments 
of  three  promotion  groups  in  the  eighth  grade — the  fast,  the  regular, 
and  the  slow  group;  (2)  to  determine  differences  among  these  same 
pupils  when  regrouped  on  the  basis  of  age,  for  the  purpose  of  finding 
out  whether  or  not  the  age  or  the  promotion  factor  is  the  more 
important;  (3)  to  determine  differences  in  arithmetical  attainments 
of  three  other  promotion  groups  in  the  seventh  grade,  "regular" 
pupils  (those  making  normal  progress),  "irregular"  pupils  (those 
repeating  because  of  transfer  of  schools,  sickness,  etc.),  and  failures 
(those  repeating  because  of  failure  to  do  the  work  of  the  grade); 
and  (4)  to  determine  differences  in  arithmetical  attainments  of  two 
more  promotion  groups  in  the  eighth  grade,  the  one  composed  of 
pupils  failing  below  the  sixth  grade,  the  other  of  those  failing  above 
the  fifth  grade. 

METHOD 

This  study  is  based  entirely  on  records  made  by  children  in  the 
Grand  Rapids  schools.  An  examination  of  the  arithmetic  folder 
used  in  the  survey  of  that  city  shows  that  on  the  front  page  of  the 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    103 

folder  the  following  question  was  asked  of  the  pupil  taking  the  test: 
"Have  you  ever  repeated  the  arithmetic  of  a  grade  because  of  non- 
promotion  or  transfer  from  other  school  ?    If  so,  name  grade 

Explain  cause."  In  addition  to  answers  to  this  question,  through 
special  request  the  children  from  a  number  of  the  schools  indicated 
the  fact  whether  they  had  ever  skipped  one  or  more  grades.  Thus 
records  were  secured  showing  normal  progress,  progress  below  nor- 
mal, and  progress  above  normal,  and  the  cause  of  slow  progress — 
repeating — ^was  ascertained  in  the  cases  where  it  occurred. 

In  comparing  the  fast,  regular,  and  slow  pupils  the  method  of 
selecting  records  of  pupils  employed  in  connection  with  the  study 
of  the  age  groups  was  adopted  for  the  purpose  of  eliminating  the 
factors  of  differences  in  giving  the  test,  and  differences  in  training. 
The  same  method  was  used  in  selecting  records  for  the  other  pro- 
motion groups.  In  this  way  150  records  were  secured  for  the  first 
division  of  the  study,  representing  50  "fast"  pupils,  50  "regular" 
pupOs,  and  50  "slow"  pupils.  For  the  second  division  of  the  study 
these  same  150  records  were  used,  merely  being  grouped  in  a  differ- 
ent way.  For  the  third  division  of  the  study,  162  records  were 
secured,  54  for  each  group.  The  data  for  the  last  part  of  the  study 
were  the  most  meager,  for  it  was  possible  to  obtain  only  32  records 
for  each  of  the  groups. 

The  records  thus  selected  were  carefully  scored;  the  attempts 
and  rights  were  tabulated;  and  average  attempts,  average  rights, 
and  percentages  of  accuracy  were  computed. 

RESULTS 

As  already  stated,  the  first  division  of  this  study  relates  to  three 
promotion  groups,  the  first  designated  as  the  "fast"  group,  com- 
posed of  pupils  who  have  skipped  one  or  more  grades;  the  second 
designated  as  the  "regular"  group,  made  up  of  pupils  who  have 
neither  skipped  nor  repeated;  and  the  third  designated  as  the 
"slow"  group,  composed  of  pupils  who  have  repeated  one 
or  more  grades.  Each  group  is  represented  by  50  pupils  in 
Grade  8-2. 

The  average  number  of  examples  worked  correctly  in  each  set 
by  the  pupils  in  each  of  these  groups  is  shown  in  Table  XL VIII. 


I04        STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

The  total  number  of  "units"  made  by  each  of  the  groups,  as  deter- 
mined by  the  system  of  weights,  is  also  presented  in  this  table.  In 
general,  the  differences  found  here  are  of  the  same  order  as  those 
discovered  in  the  study  of  the  age  groups.  The  "fast"  pupils  are 
decidedly  superior  to  the  "slow"  pupils,  while  the  "regular"  pupils 
occupy  an  intermediate  position.  These  differences  are  more 
marked  in  the  more  complex  than  in  the  simpler  examples. 


TABLE  XLVIII 

Average  "Rights"  Made  in  Each  Set  by  Three  Promotion  Groups. 
Data  from  150  Pupils 


Grade  8-2. 


Group 

Set 

t 

A 

a 

c 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

Fast 

30.6 
29.9 
30.2 

251 
24.1 
26.1 

21. 1 
20.7 
20.6 

22.7 
22.8 
22.4 

7-7 
7.8 

7-2 

10.9 
10. 1 
10.3 

6.8 
6.7 
S-9 

8.4 
8.5 
8.1 

41 
4-1 
3-7 

5-9 
5-3 
S-4 

10.7 
10.7 
10.2 

4-9 

4-4 
41 

5-8 
51 

5-2 

2-3 

1.8 
1-7 

6.6 
5-3 

4-7 

442 
414 
399 

Regular 

Slow 

The  facts  for  the  "attempts,"  made  by  the  same  groups,  appear 
in  Table  XLIX.  Although  the  differences  between  the  groups  are 
not  quite  so  marked  here  as  in  the  case  of  the  "rights,"  they  are 
very  substantial,  being  larger  than  those  found  in  the  study  of  the 
Cleveland  age  group. 

TABLE  XLIX 

Average  "Attempts"  Made  in  Each  Set  by  Three  Promotion  Groups. 
Grade  8-2.    Data  from  150  Pupils 


Set 

^§ 

Group 

A 

B 

c 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

Fast 

Regular. . . 
Slow 

30.7 
30.2 

30s 

254 
24.4 
26.5 

22.2 
21.7 
22.1 

23-4 
233 
23.1 

8.2 
7.6 
7-7 

II. 7 
10.8 
II. 4 

7.6 
75 
6.9 

11. 4 
10.9 
11. 4 

4-9 
4-9 
4-9 

7-5 
7.0 

6.9 

10.9 
II. 0 
10.4 

6.6 

6.2 
5-7 

7.2 
6.7 
6.7 

2.9 
2.9 

2.7 

9.1 
8.2 
7-9 

S16 
492 
487 

The  average  accuracy  achieved  by  the  three  groups  in  all  the 
sets  is  shown  in  Table  L.  As  would  be  suspected  from  the  facts 
presented  concerning  "rights"  and  "attempts,"  the  "fast"  group 
is  the  most  accurate. 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS     105 

In  order  to  discover  the  relative  importance  of  the  factors  of 
promotion  and  of  age,  the  150  pupils  used  in  the  study  just  dis^ 
cussed  were  regrouped  on  the  basis  of  age,  the  50  youngest  being 
placed  in  one  group,  the  50  oldest  in  a  second  group,  and  the  remain- 
ing 50  in  a  third  group. 

TABLE  L 

Average  Accuracy  in  All  Sets.    Three  Pro- 
motion Groups,    Grade  8-2 


Fast.... 
Regular 
Slow.  .  . 


Accuracy 


85.7 
84.2 
81.9 


Tables  LI,  LII,  and  LIII  are  identical  in  form  with  the  three 
tables  just  discussed  in  connection  with  the  age  groups,  the  first 
presenting  the  facts  for  the  "rights,"  the  second  those  for  the 
"attempts,"  and  the  third  those  for  the  accuracy  of  each  of  these 
three  age  groups.     The  very  interesting  and  surprising  fact  brought 


TABLE  LI 

Average  "Rights"  Made  in  Each  Set  by  Three  Age  Groups.    Grade  8-2. 
Data  from  150  Pupils 


Set 

Groxjp 

A 

B 

c 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

Young 

Normal 

Old 

311 
309 
28.7 

26.8 
23.6 
22.7 

20.9 
21.6 
20.0 

24.2 
22.6 
21.2 

8.0 
7-7 
6.9 

II. I 

10.3 

9.9 

71 
6.8 
5-6 

9.1 
8.8 
6.9 

4-3 
4-3 
3-3 

5.6 
6.1 
4-9 

II. 2 

10.6 

9-7 

4-9 
4-7 
3-6 

5-6 
5-7 
4.8 

2-3 

2.0 
1.6 

6.8 

5-8 
4.2 

452 
431 
369 

out  by  these  tables  on  the  promotion  groups  is  that  the  differences 
between  the  "young"  and  the  "old"  groups  are  greater  than  the 
differences  between  the  "fast"  and  the  "slow"  groups.  That  is, 
the  younger  pupils  of  the  grade  are  more  superior  to  the  older  pupils 
of  the  grade  than  the  "fast"  pupils  are  to  the  "slow"  pupils.  Of 
course  the  data  do  not  warrant  any  definite  conclusions,  but  the 
indications  are  that,  so  far  as  attainments  in  arithmetic  are 
concerned,  pupils  may  fail  of  promotion,  or  rather  may  not  be 


io6        STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


recommended  for  extra  promotion,  because  they  are  thought  to 
be  too  young  rather  than  because  of  their  inability  to  do  more 
advanced  work. 

TABLE  LII 


Average  "Attempts"  Made  in  Each  Set  by  Three  Age  Groups. 
Data  from  150  Pupils 


Grade  8-2. 


Set 

r 

A 

B 

c 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

Young 

Normal.  .  .  . 
Old 

313 
31-3 
28.9 

27.0 
24.0 
23.1 

21.7 
22.8 
21.5 

24.7 
233 
21.7 

8.3 
8.2 

7-3 

II. 8 
II. I 
10.9 

7.8 
7.6 
6.5 

11. 6 

11. 7 
10.3 

S-2 

4-3 

7-7 
7-4 
6.4 

II. 4 

II. 0 

9.9 

6.8 
6.3 
5-4 

71 
7.2 
6.4 

31 
2.9 

2.6 

9.0 

8.7 
7-7 

524 

5" 

457 

In  Grade  7-2,  108  pupils  who  had  repeated  one  or  more  grades 
were  divided  into  two  groups,  the  one  being  made  up  of  54  pupils 
who  had  repeated  because  of  sickness,  or  transfer  of  school,  etc.,  the 
other,  of  54  pupils  who  had  repeated  because  of  failure  to  do  the 
work  of  the  grade.  A  control  group  of  54  pupils  making  normal 
progress  was  also  used  for  the  study. 

TABLE  LHI 

Average  Accuracy  in  All  Sets.    Three  Age 
Groups.    Grade  8-2 


Young. 
Normal 
Old. . . , 


Accuracy 


86.3 

84.3 
80.7 


The  average  number  of  "rights"  made  in  each  set  of  the  test 
by  each  of  the  groups  is  found  in  Table  LIV,  together  with  a  state- 


TABLE  LIV 

Average  "Rights"  Made  in  Each  Set  by  Three  Promotion  Groups. 

Data  from  162  Pupils 


Grade  7-2. 


Set 

'^V 

Group 

A 

B 

C 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

r 

Regular 

Irregular 

Failures 

30.2 

293 
28.6 

22.1 
22.4 
20.2 

19.2 
21.0 
19.2 

20.8 
20.2 
193 

7-4 
7-4 
6.8 

9.8 

8.5 
8.0 

6.0 
5-8 
5-4 

9.0 
9.8 
8.8 

3-7 
2.6 

2-5 

7.2 

5-7 
4.6 

8.5 
8.3 
7.6 

4.2 
4-2 
3-7 

4-9 
4-7 
4.0 

1.2 
I.I 

5-4 
4.8 
4.2 

400 
376 
342 

ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    107 

ment  of  total  scores.  An  examination  of  the  table  shows  quite  con- 
siderable differences  between  the  groups.  The  "irregular"  pupils 
are  superior  to  the  "failures,"  and  the  "regular"  pupils  are  superior 
to  the  "irregular."  These  differences  are  also  more  marked  in 
examples  of  the  more  complex  than  in  those  of  the  simpler  type. 
Table  LV,  the  table  of  "attempts,"  shows  differences  of  the  same 

TABLE  LV 

Average  "Attempts"  Made  in  Each  Set  by  Three  Promotion  Groups. 
Grade  7-2.    Data  from  162  Pupils 


Gkotip 

Set 

A 

B 

c 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

1= 

Regular 

Irregular.  . . . 
Failures 

303 
29.6 
29.0 

22.4 
22.7 
20.6 

20.4 
21.9 

20.7 

21. 1 
20.7 
19.9 

7-7 
7-9 
7.2 

10.2 

9-3 
8.8 

6.6 
6.5 
6.3 

II. 7 
12.1 
12.0 

4-S 
3-4 
3-6 

8.4 
6.9 
5-9 

9.0 

8.5 
8.1 

5-6 

5-2 

5-1 

6.1 
5-7 
5-6 

2.3 
2.0 
2.0 

8.6 

8.3 
8.0 

472 
446 
427 

order.  One  difference,  however,  should  be  noted,  and  that  is  that 
the  "irregular"  pupils  are  more  inferior  to  the  "regular"  pupils  in 
"attempts"  than  in  "rights."  In  other  words,  the  difference 
between  these  two  groups,  as  borne  out  by  Table  LVI,  is  not  due 

TABLE  LVI 

Average  Accuracy  in  All  Sets.    Three  Pro- 
motion Groups.    Grade  7-2 


Regular. 
Irregular 
Failures . 


Accuracy 


84.8 

843 
80.1 


to  inaccuracy  on  the  part  of  the  "irregular"  pupils,  but  rather  to 
slowness  or  timidity.  A  graphical  comparison  of  the  three  groups 
is  found  in  Diagram  37. 

Now  the  difference  between  the  "failures"  and  the  "irregular" 
pupils,  which  is  in  favor  of  the  latter,  is  not  at  all  surprising,  but 
why  should  the  "regular"  pupils  be  superior  to  the  "irregular" 
pupils  ?  The  latter  have  repeated  the  work  of  a  grade  because  of 
sickness  or  transfer  of  schools  and  not  because  of  inability  to  do  the 


io8        STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

work.  In  other  words,  they  should  not  naturally  be  markedly 
different  from  the  "regular"  pupils,  for  any  pupil  may  be  sick  and 
any  pupil  may  be  transferred  from  one  school  to  another.  Further- 
more, it  would  seem  that,  if  the  repeating  of  a  grade  is  a  good  thing 
for  a  pupil,  the  "irregular"  pupil  in  a  particular  grade  should  be 
superior  to  the  "regular"  pupil  of  the  same  grade,  since  he  has  had 


TR^yti  la  1^    

Irregular'  — —  •— - 
Faifures    —  —  — - 


Diagram  37. — Average  "units"  made  in  each  set  by  three  promotion  groups  in 
Grade  7-2. 

a  year's,  or  a  portion  of  a  year's,  extra  training.  Since  such  is  not 
the  case,  it  must  be  that  the  repeating  of  the  grade  is  not  so  valuable 
an  experience  to  the  pupil  as  some  have  been  led  to  believe.  From 
the  evidence  here  set  forth — inconclusive  of  course  because  of  its 
paucity — the  indications  are  that  the  repeating  of  a  grade  for  what- 
ever cause  reacts  upon  the  child  and  may  even  become  a  cause  of 
failure. 

The  third  study  of  promotion  groups  is  based  on  the  facts  pre- 
sented in  Tables  LVII,  LVIII,  and  LIX.  In  the  eighth  grade  64 
pupils  who  had  repeated  one  or  more  grades  were  divided  into  two 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    109 

groups,  the  one  being  composed  of  32  pupils  who  had  repeated  below 
the  sixth  grade,  the  other,  of  32  pupils  who  had  repeated  above  the 
fifth  grade. 


TABLE  LVII 

Average  "Rights"  Made  in  Each  Set  by  Two  Promotion  Groups.    Grade  8-2. 

Data  from  64  Pupils 

Set 

**  @ 

Gkoup  Fahjdig 

<  S 

A 

B 

C 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

r 

Below  sixth 

grade 

Above  fifth 

30.8 

234 

20.1 

21.7 

7-4 

10.8 

5-8 

8.1 

3-9 

5-7 

10.7 

4.6 

4-9 

1-7 

4.8 

403 

grade 

30.0 

234 

21.5 

23.1 

7.6 

9.9 

S-9 

7.8 

3-3 

5-6 

9.2 

4.0 

S-3 

i-S 

4-3 

387 

The  tables  show  slight  dilBferences  in  favor  of  the  group  failing 
below  the  sixth  grade,  but  since  the  differences  are  so  slight  and  the 

TABLE  LVIII 
Average  "Attempts"  Made  in  Each  Set  by  Two  Promotion  Groups. 


Grade  8- 

-2, 

Data  from  64  Pupils 

Group  Failing 

Set 

2.* 

A 

B 

C 

D 

£ 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

r 

Below  sixth 
grade. . . . 

Above  fifth 
grade. .  . . 

31-3 
304 

23 -9 
23.8 

22.8 
22.6 

22.3 
23.8 

8.2 

8.0 

II. 9 
10.8 

71 
6.9 

II. 4 
II. 4 

4-4 

7-4 
71 

n.o 
9.6 

6.x 

5-7 

6.8 
6.7 

2.6 

2.6 

7-5 
8.1 

493 

480 

cases  so  few,  whatever  conclusions  are  drawn  must  be  wholly 
tentative.     Two   explanations   of   the   differences   suggest  them- 


TABLE  LIX 

Average  Accuracy  in  All  Sets.    Two  Promo- 
tion Groups.    Grade  8-2 


Failing  below  sixth  grade. 
Failing  above  fifth  grade. 


Accuracy 


81.7 
80.6 


selves:  first,  it  may  be  that  the  pupil  has  more  or  less  recovered 
from  the  failure  in  the  earlier  grades  by  the  time  he  has  reached 


no        STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

the  eighth  grade;  second,  it  may  be  that  failure  in  the  lower 
grades  is  due  to  causes  somewhat  different  from  those  that  oper- 
ate in  the  upper  grades.  In  the  former  the  pupU  may  be  unable 
to  realize  what  he  is  attending  school  for,  and  may  therefore 
be  quite  indifferent  to  failure.  That  is,  he  may  fail  because  of 
carelessness  rather  than  because  of  inability.  In  the  upper 
grades,  on  the  other  hand,  failure  may  more  often  be  the  result 
of  inability. 

SUMMARY 

1.  With  reference  to  the  number  of  examples  worked  correctly 
by  the  pupils  in  the  four  age  groups,  it  may  be  said  that  on  the 
average  the  younger  groups  are  superior  to  the  older  groups;  that 
this  superiority  is  more  marked  in  the  later  than  in  the  earlier 
grades;  and  that  it  is  also  more  marked  in  the  handling  of  the  more 
complex  than  in  the  handling  of  the  simpler  types  of  examples. 

2.  In  the  number  of  examples  attempted  the  study  reveals  no 
clear  differences  between  any  two  of  the  four  age  groups. 

3.  On  the  average  the  younger  pupils  are  found  to  be  more 
accurate  in  their  work  than  the  older  pupils;  these  differences  are 
on  the  whole  quite  uniform  from  grade  to  grade;  and  they  are  more 
pronounced  in  the  more  complex  than  in  the  simpler  examples. 

4.  A  study  of  "fast,"  "regular,"  and  "slow"  pupils,  as  deter- 
mined by  promotion  facts,  reveals  differences  of  the  same  order  as 
those  just  stated  concerning  the  age  groups,  the  "fast"  correspond- 
ing to  the  "young"  and  the  "slow"  to  the  "old." 

5.  A  regrouping  of  these  same  pupils  ("fast,"  "regular,"  and 
"slow")  on  the  basis  of  age  shows  the  differences  to  be  more  pro- 
nounced than  when  the  pupils  are  grouped  according  to  the  rate  of 
promotion. 

6.  This  last  statement  would  indicate  a  tendency  to  keep  pupils 
in  a  grade  because  of  youth. 

7.  "Failures"  (pupils  repeating  because  of  inability  to  do  the 
work  of  the  grade)  are  inferior  to  "irregular"  pupils  (pupils  repeat- 
ing because  of  sickness,  transfer  of  school,  etc.),  and  the  latter  are 
inferior  to  "regular"  pupils  (pupils  making  just  normal  progress). 


ABILITIES  OF  CERTAIN  AGE  AND  PROMOTION  GROUPS    iii 

The  relation  between  the  two  latter  groups  is  of  significance  as 
indicating  the  injurious  effects  of  repeating, 

8.  The  evidence,  inconclusive  because  of  the  small  number  of 
cases  involved,  indicates  that  among  eighth-grade  pupils  the  group 
made  up  of  pupils  who  had  failed  below  the  sixth  grade  is  superior 
to  the  group  composed  of  pupils  who  had  failed  above  the  fifth 
grade. 


CHAPTER  VI 

A  COMPARISON  OF  THE  ARITHMETICAL  ABILITIES  OF 
CERTAIN  RACE  GROUPS 

The  object  of  the  present  study  is  to  make  a  comparison  of  the 
arithmetical  abilities  or  attainments  of  the  children  of  five  races, 
viz.,  American,  Hollander,  German,  Swede,  and  Slav,  with  a 
view  (i)  to  determine  whether  or  not  there  are  racial  differences, 
and  (2)  to  discover  the  nature  and  extent  of  the  differences,  if  such 
are  found  to  exist. 

METHOD 

When  the  arithmetic  test  was  given  in  the  Grand  Rapids  schools, 
the  principals  of  those  schools  in  which  several  races  were  repre- 
sented in  the  school  population  were  requested  to  have  the  teachers 
indicate  on  the  test  sheet  the  race  of  each  pupil  taking  the  test  in 
the  upper  grades.  Mixed  schools — that  is,  schools  in  which  several 
races  were  represented — were  chosen  because  it  was  thought  that 
the  comparison  of  races  should  be  made  on  the  basis  of  the  records 
of  individuals  who  had  been  subjected  to  the  same  school  influences. 
It  was  considered  that  this  would  be  a  more  valid  method  than  the 
comparison  of  average  records  made  by  schools  in  which  the  various 
races  were  predominant,  since  such  differences  as  would  be  found 
might  very  likely  be  due  to  differences  in  training. 

The  principals  were  told  to  call  a  pupil  an  "American"  if  both 
parents  were  born  in  this  country.  Otherwise  the  race  of  the  pupil 
was  to  be  indicated  as  that  of  the  parents.  Thus,  if  both  parents 
were  bom  in  Holland  the  pupil  was  called  a  "Hollander";  if  one 
was  born  in  Holland  and  the  other  in  Germany,  the  pupil  was  called 
"Hollander- German,"  and  so  on. 

In  answer  to  this  request  made  of  the  principals,  returns  were 
secured  from  at  least  one  of  the  upper  grades  in  eleven  schools. 
These  records  were  examined  and  classified  on  the  basis  of  race. 
If  there  was  any  doubt  about  the  race  of  the  pupil,  the  record  was 
not  used.    For  instance,  if  a  pupil  with  a  name  like  "Putowski" 


ABILITIES  OF  CERTAIN  RACE  GROUPS 


"3 


gave  his  race  as  American,  the  record  was  discarded.  The  records 
made  by  pupils  of  mixed  parentage  were  also  rejected.  The  group 
"Slavs"  is  composed  of  Russians  and  Poles  with  a  few  Lithuanians 
and  Bohemians;  the  "Swedes"  include  a  few  Norwegians  and 
Danes;  the  other  races  are  as  the  terms  would  indicate. 

In  Table  LX  appears  a  statement  of  the  number  of  pupils  of 
each  of  the  five  races  whose  records  were  finally  selected  for  use  in 
the  study.  The  table  also  shows  the  distribution  of  the  pupils  of 
each  race  through  the  five  upper  grades,  6-2  to  8-2  inclusive,  to 
which  the  study  is  confined.  It  will  be  noted  that  the  Americans 
and  Hollanders  are  well  and  about  equally  represented.  The  Slavs 
in  Grades  8-1  and  8-2  and  the  Swedes  in  Grades  7-2  and  8-2  are 
especially  poorly  represented.  These  facts  must  be  borne  in  mind 
when  the  results  are  interpreted. 

TABLE  LX 

Number  of  Pupils  of  Each  Race-  in  Each  Grade  Whose  Records  Were  Used 

IN  This  Study 


Race 

Grade 

Total 

6-2 

7-1 

7-2 

8-1 

8-2 

American 

60 
47 
25 
16 

19 

36 
49 
31 
15 
14 

Si 
55 
23 
10 

17 

SI 
S6 
17 
16 
8 

50 
50 
21 
10 
8 

250 

257 

117 

67 

66 

Hollander 

German 

Swede 

Slav 

Total 

167 

145 

158 

148 

139 

757 

The  distribution  of  the  pupils  of  these  five  races  through  the 
different  schools  and  the  composition  of  each  class  from  which 
records  were  taken  are  presented  in  Table  LXI.  This  table  will  be 
understood  if  read  in  this  way:  In  the  Coldbrook  Elementary 
School  the  class  of  Grade  6-2  was  composed  of  24  pupils  of  whom 
5  were  clearly  Americans,  2  Hollanders,  2  Germans,  i  a  Swede, 
4  Slavs,  and  10  members  of  other,  uncertain,  or  mixed  races.  The 
table  is  read  in  the  same  way  for  each  of  the  grades  and  for  each  of 
the  schools. 

The  method  used  in  working  up  the  data  should  now  be 
explained  in  detail  because  of  its  complex  character.    If  these  five 


114        STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


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ABILITIES  OF  CERTAIN  RACE  GROUPS  115 

races  had  equal  representation  in  each  of  the  classes  from  which 
records  have  been  taken,  or  if  we  had  sufficient  data  so  that  equal 
numbers  of  the  races  might  be  taken  from  each  class,  as  was  done 
in  the  study  of  age  groups,  the  question  would  be  a  very  simple  one. 
But,  since  the  races  are  not  equally  represented  in  the  classes  and 
the  data  are  strictly  limited,  some  method  must  be  found  of  elimi- 
nating differences  of  school  training  and  methods  of  giving  the  test. 
Referring  to  Table  LXI  again,  let  us  consider  the  problem  as  related 
to  the  records  of  the  pupils  in  Grade  8-2,  and  for  the  moment  let  us 
confine  ourselves  to  the  records  made  by  the  Americans  and  the 
Hollanders.  Five  of  the  American  pupils  in  this  grade  are  taken 
from  the  Coldbrook  School,  while  but  2  Hollanders  are  taken  from 
the  same  school.  Now  if  the  training  received  in  the  Coldbrook  is 
superior  to  that  received  in  the  other  schools,  and  if  the  records 
made  by  the  50  American  pupils  and  the  records  made  by  the  50 
Hollanders  in  the  grade  be  averaged  without  eliminating  this  factor 
of  difference  of  training,  the  American  group  would  be  given  an 
advantage  because  of  its  greater  representation  in  a  superior  school. 
The  method  adopted  for  eliminating  differences  in  school  train- 
ing is  as  follows:  The  records  made  by  the  entire  grade  in  one 
school  are  taken  as  a  base,  and  a  system  of  coefficients  is  computed 
by  the  use  of  which  the  records  made  in  the  other  schools  may  be 
converted  into  the  records  of  the  school  taken  as  a  base.  Thus  the 
thing  that  is  actually  done  is  to  determine  what  the  records  of  the 
different  pupils  and  groups  of  pupils  would  have  been  if  they  had 
all  been  in  the  same  school,  subjected  to  the  same  school  influences. 
To  be  more  concrete,  let  us  again  turn  to  Table  LXI  and  indicate 
specifically  how  the  method  is  applied  in  dealing  with  the  race 
groups  in  Grade  8-2.  The  table  shows  that  the  data  have  been 
taken  from  Grade  8-2  of  six  schools.  The  next  thing  that  must  be 
done  is  to  find  the  average  records  made  by  these  six  grades  in  each 
of  the  15  sets  of  the  test.  These  facts  are  presented  in  Table  LXII. 
Now,  taking  the  average  score  made  in  the  Union  School  as  a  base, 
a  coefficient  is  determined  for  each  set  in  each  school  by  dividing 
the  record  made  in  the  Union  School  by  the  record  made  in  each 
of  the  other  schools.  Thus  a  set  of  coefficients  is  found  by  which 
the  records  of  the  five  schools  may  be  converted  into  the  records 


ii6        STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


made  by  the  Union  School,  which  means  that  differences  in  school 
training  and  differences  in  giving  the  test  are  overcome.  By  using 
the  set  of  coefficients  for  any  one  of  the  schools,  it  is  possible  to 

TABLE  LXII 
Average  "Rights"  Made  in  Each  Set  by  Grade  8-2  in  Six  Schools 


B 


D 


H 


K 


M 


N 


O 


Union 

Coldbrook 

Diamond 

East  Leonard.. . 

Lexington 

South  Division. 


28.3 
28.8 

317 
34-8 
32.0 
32.5 


23s 
28.8 

25-5 
30.8 
30.8 
26.5 


16.8 
22.3 
20.7 

21-5 

23s 

20.5 


19.0 
27.2 
22.7 

24.8 

24.6 

23.0 


9-3 
13s 
10.3 
12.9 
12.3 
10. o 


S-i 
8.8 
12.0 
14.2 
7.8 
8.5 


4-7 
S-S 
4-7 
6.7 
4-7 
4.0 


9-4 
12.2 
10.3 

13  I 
II. o 
10.3 


4-9 
6.5 
4-9 
6.1 
5-8 
4-3 


1.9 
31 
2-3 
41 
2.7 


4.8 
5-3 
4-3 
6.1 
6.0 
4.0 


determine  what  sort  of  a  record  a  group  of  pupils  in  that  school 
would  have  made  if  they  had  been  trained  and  tested  in  the  Union 
School.     These  coefficients  are  presented  in  Table  LXIII. 

TABLE  LXIII 

Value  of  Each  Example  in  Each  Set  for  Grade  8-2  in  Each  of  Six  Schools 
IN  Terms  of  Record  Made  by  Grade  8-2  in  Union 


A 

B 

c 

D 

E 

F 

G 

H 

I 

J 

K 

L 

M 

N 

0 

Union 

1. 00 

1. 00 

1. 00 

1. 00 

1. 00 

1. 00 

1. 00 

1. 00 

1. 00 

1. 00 

1. 00 

1. 00 

1. 00 

I.OO 

1. 00 

Coldbrook.. 
Diamond.. . 
East 

0.98 
0.89 

0.82 
0.92 

0.75 
0.81 

0.70 
0.84 

0.90 
0.94 

0.69 
0.90 

0.85 
I  05 

0.58 
0.43 

0.85 
1. 00 

0.79 
0-93 

0.77 
0.91 

0-7S 
1. 00 

0.72 
0.98 

0.63 
0.83 

0.91 
1. 12 

Leonard. . 
Lexington.  . 
South 

0.81 
0.88 

0.76 
0.76 

0.78 
0.72 

0.77 
0.77 

0.84 
0.83 

0.72 
0.76 

0.72 
0.87 

0.36 
0.65 

0.70 
1. 00 

0.78 
0.81 

0.72 
0.8s 

0.80 
0.84 

0.74 
0.7s 

0.46 
0.70 

0.79 
0.80 

Division. . 

0.88 

0.89 

0.82 

0.83 

0.99 

0.93 

1.06 

0.60 

1. 18 

1.08 

0.91 

1. 14 

0.92 

1.06 

1.20 

All  that  remains  now  is  merely  to  point  out  the  way  in  which 
these  coefficients  are  applied  to  the  race  groups.  Referring  again 
to  the  section  of  Table  LXI  which  presents  the  facts  for  Grade  8-2, 
we  find  5  American  pupils  in  the  Coldbrook  School.  The  total 
number  of  examples  worked  correctly  in  each  of  the  sets  by  these 
5  pupils  is  determined.  Turning  to  Table  LXIII  we  find  the 
Coldbrook  coefficient  for  Set  A  to  be  0.98.  The  total  number  of 
"rights"  made  by  the  5  pupils  is  multiplied  by  this  quantity.  The 
same  thing  is  done  for  each  of  the  other  sets.  Then  we  pass  to  the 
4  pupils  in  the  Diamond  School  and  repeat  the  process,  and  like- 


ABILITIES  OF  CERTAIN  RACE  GROUPS  117 

wise  with  the  Americans  in  each  of  the  other  schools.  Then  the 
total  scores  thus  made  in  each  of  the  sets  by  the  American  pupils 
in  the  six  schools  are  added.  This  grand  total  is  then  divided  by 
50,  since  that  is  the  entire  number  of  American  pupils  in  the  grade, 
and  an  average  score  is  secured  for  each  of  the  sets.  Like  procedure 
is  followed  for  each  of  the  other  four  race  groups. 

An  objection  which  may  be  raised  to  this  method  is  that  in 
eliminating  differences  due  to  training  and  to  the  giving  of  the  test 
race  differences  are  also  eliminated.  The  answer  to  this  objection 
is  that  it  would  be  valid  if  mixed  schools — that  is,  schools  in  which 
several  races  are  represented — were  not  used  for  the  study.  To 
the  extent  that  a  difference  between  the  average  records  made  by 
two  schools  is  due  to  the  presence  of  an  inferior  or  a  superior  race 
group  in  one  of  the  schools,  the  method  does  eliminate  race  differ- 
ences as  well  as  differences  due  to  school  training  and  methods 
of  giving  the  test.  But,  since  the  schools  are  mixed,  no  one  race 
can  greatly  affect  the  average  score  of  a  school.  The  objection  thus 
resolves  itself  into  the  question  of  the  racial  composition  of  the 
classes  from  which  the  records  used  were  taken.  We  must  there- 
fore refer  again  to  Table  LXI,  in  which  the  composition  of  each  of 
these  classes  is  given  in  detail.  If  it  be  remembered  that  the  cate- 
gory of  "others"  includes  representatives  of  races  other  than  the 
five  used  in  this  study,  pupils  of  mixed  parentage,  and  pupils  con- 
cerning whom  there  was  some  doubt  as  to  race,  the  table  shows 
that  in  no  single  instance  does  one  race  represent  a  majority  of  the 
class,  and  in  only  two  cases.  Diamond,  Grades  6-2  and  7-2,  does 
one  approach  representing  a  majority.  Thus  it  is  quite  evident 
that  the  differences  between  any  two  schools  cannot  be  attributed 
in  any  appreciable  degree  to  the  predominance  of  any  race  in  one 
or  the  other  of  the  schools.  It  may  be  that  the  method  used  in  this 
study  does  minimize  to  some  small  extent  racial  differences,  but  to 
a  very  small  extent  if  at  all. 

RESULTS 

The  average  scores  made  in  each  set  as  computed  by  the  method 
just  outlined,  by  each  of  the  five  race  groups  in  Grades  6-2  to  8-2, 
are  found  in  Table  LXIV.    An  examination  of  the  table  seems  to 


ii8 


STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 


n 


(K4 


SA^S 


S3p3MS 


streoMSQ 


sjapirejlOH 


sveouscny 


SABIS 


SSpSMg 


sncousQ 


SJSpiTBlIOH 


sa«3U3uiy 


SABIS 


sapsMg 


EirenusQ 


sisprreiioH 


sxre3U3aiY 


SABIS 


sapaMg 


sireaijsr) 


sjaptrciiojj 


sa«3U3aiy 


SABI5 


sspaMg 


sncouaf) 


SJsptreiioQ 


saeausuiY 


VO  i«  ■^  0>  M  vO   v>  o>vO  00   0>  *  fj  O   w 


00    Tj-  Ov  w   f^OO   lOPOt^M    ^M    «00    O 


■^  t>.00   t^  H   w   t^  O    O    O    POOO  00   lO  ■* 


00Ov«5i-it>"t^>H00NO.i-ic*iOTl-p»5 


't  'J-OO   fOl>-0    cOTj-roo^O    cO'^toO 


0<  M    OiOO  00  t^l^O   cOtOM   itO\O\f0 


vO   cO"*i-i  t^O^OvO\f'30^«or^»OM   t^ 


O   O*  lOOO   M   tH    M   t^vo    w    fO  M   CO  r^  <s 


Tj-  •«*■  O^  P<   l^  t^  0>  W)00  m  lo  vo  O    O^  PI 
N    <S    11    <M 


c<oO'^>'>>oOt^i-<«'JMOOMO^ 


00   O*  0>  i-i  00   M 


low     MO     flMOO     M 


t^wwMUiNrOi-ifOOt^Ot^Tl-M 


M   O^"))-*   0>00^f00   'J"0»'i"M   o>o> 


t^M     IHt^<N     MVOOOOO     O^fO"!   fOVO     M 


lO  coo  00   M    H    CO  O*  O^OO   0\  O*  t-i   lO  01 


O   ^  M    O    CO  t^O    O^O   *>•  tJ-  c>«   CO  O^  I 


00  «oo  o>  »o  -"t  p)  >noo  CO  w  i^  O>oo  r^ 


M    O    O   lo  O   0*00  00  OO    CO  t^  VCO    w    CO 


OOOOOiOMi-iiHt^.'"*-  ■^O   lO  t^  M 


•*«Ol^O    w   "itMOOOO   TtoO   r~  O*  "^ 


lo  t^  T^  CO  « 


lo  0\0   CO  «  00  00   o\oo  o 
00   tOO^co^l^cO'^M   lo 


M    MOO    <>»    "tO^M    P400OO00    ■^O  00 
t~.0  00     CO'^t^CO^M     IT) 


l^  0\  t>.  «  o 


«  O   t^  O   0\  O   «O00    COOO 
l^ir>i^cOcol>»co^M    CO 


M    w   «    O   M 


l>.0    t^M    Iv.t^'^CON    O 

t^>OCOcocor^cO'<J"M   vo 


O    CO  "l  c«   rf 


«J-  Tt  O    O*  coo    CI    Ci    CO  M 
l>»TfO*M     "*00CO'tM     m 


<P3UQWfadWwt-^WHjS!z;d 


ABILITIES  OF  CERTAIN  RACE  GROUPS 


119 


TABLE  LXV 

Av£HAGE  Number  of  "Units"  Made  in  All  Sets,  except  Sets  H  and  0  (Frac- 
tions), BY  Each  of  Five  Races  in  Grades  6-2  to  8-2 


Race 


Grade 

6-2 

7-1 

7-2 

8-1 

8-2 

279 

301 

295 

336 

320 

I, S3 1 

278 

330 

302 

338 

310 

i,SS8 

283 

303 

328 

347 

326 

1,587 

304 

321 

299 

358 

340 

1,622 

316 

338 

339 

39.0 

317 

1,700 

Average 


American . 
Hollander 
German . . 

Swede 

Slav 


306 
312 
317 
324 
340 


indicate  that  there  are  some  differences  and  that  they  are  in  favor 
of  the  Swedes  and  Slavs.  But  since  the  differences  do  not  appear 
to  be  consistently  in  any  one  set,  Table  LXV  is  presented,  in  which 

SIglv        >-— —  —     HoUanc/er    

Si^a</e.       Ametican    — — — 

O  St  man 


J90 

340 
320 
300 

260 
2^0 
220 
200 
f80 
160 
MO 
120 

too 

80 
M 
40 

20 


,^\ 


s-a 


7'/ 


«■/ 


8--? 


7-2 
Diagram  38. — A  comparison  of  records  made  by  five  race  groups 


I20        STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

appear  the  total  scores  made  by  each  of  the  races  in  the  five  grades. 
These  total  scores  are  obtained  by  applying  the  system  of  weights 
derived  in  chapter  iii  to  the  average  scores  given  in  Table  LXIV. 

Since  these  same  facts  are  presented  graphically  in  Diagrams  38 
and  39,  our  attention  may  be  turned  toward  them.  The  first  of 
these  diagrams  shows  at  a  glance  that  the  American  children  are 
not  prodigies  when  compared  with  the  children  of  the  other  races. 
With  the  exception  of  the  records  for  Grade  7-1  the  Hollanders  and 
Americans  are  in  close  agreement,  and  as  compared  with  the  other 
races  they  seem  to  be  somewhat  inferior.  The  Germans  and  the 
Swedes  are  intermediate  groups,  while  the  Slavs  really  seem  to  be 
superior.  The  curve  for  the  Slavs  becomes  doubly  significant  when 
it  is  remembered  that  they  had  the  smallest  representation  in 
Grades  8-1  and  8-2.  It  is  in  the  latter  grade  only  that  this  race 
does  not  hold  first  place. 


German  017) 
Hollar,t/erOlZ) 


Diagram  39. — A  comparison  of  average  numbers  of  units  made  in  all  sets  (Sets  H 
and  O  excluded)  by  five  race  groups — Grades  6-2  to  8-2  combined. 

CONCLUSIONS 

The  conclusions  that  may  be  drawn  from  this  study  must  of 
course  be  more  or  less  tentative  because  of  the  small  number  of 
cases  used.  It  seems  safe,  however,  to  conclude  that  the  differences 
in  arithmetical  abilities  of  children  of  American  parentage  and  chil- 
dren of  Holland  descent  are  very  small,  if  they  exist  at  all.  The 
same  may  also  be  said  of  the  German  children,  as  compared  with 
these  two  groups,  while  the  indications  are  that  the  Swedes  and  the 
Slavs,  and  especially  the  latter,  are  superior.  But  as  to  whether 
these  differences  are  due  to  the  operation  of  biological  or  social 
factors,  the  present  study  furnishes  no  evidence. 


CHAPTER  VII 
SUMMARY  AND  CONCLUSIONS 

In  this  chapter  a  very  brief  summary  of  each  of  the  studies  will 
be  made,  together  with  a  statement  in  each  case  of  the  conclusions 
that  may  be  drawn.  It  should  be  reasserted,  however,  that  these 
studies  should  all  be  supplemented  by  experimental  evidence. 

THE  NATURE  OF  THE  TEST  AND  COLLECTION  OF  DATA 

1.  The  arithmetic  test  used  in  these  studies  was  developed  for 
the  definite  purpose  of  meeting  a  need  felt  by  the  staff  of  the 
Cleveland  Survey,  which  was  not  met  by  any  existing  test. 

2.  The  test  as  devised  is  a  speed  test  which  measures  attainments 
and  indicates  weaknesses  in  the  four  fundamental  operations  and 
fractions. 

3.  In  addition  it  tests  knowledge  of  tables,  the  ability  to  add 
short  columns,  to  bridge  the  "attention  spans,"  and  to  "carry"; 
in  subtraction  it  tests  knowledge  of  the  tables  and  the  ability  to 
"borrow";  in  multiplication  it  tests  knowledge  of  the  tables, 
ability  to  "carry,"  and  ability  to  add  in  connection  with  multipli- 
cation, as  well  as  mastery  of  the  mechanics  of  working  the  more 
complex  examples  in  multiplication;  in  division  it  tests  knowledge 
of  the  tables,  ability  to  "carry"  in  short  division,  and  ability  to 
solve  two  types  of  examples  in  long  division,  the  one  involving 
neither  "carrying"  nor  borrowing  and  the  other  involving  both; 
and  it  tests  the  ability  to  apply  these  four  fundamental  operations 
to  the  working  of  examples  in  fractions. 

4.  The  test  was  given  to,  and  the  results  were  secured  from,  834 
classes  in  the  schools  of  Cleveland  and  Grand  Rapids.  In  both  cities 
the  test  was  given  almost  entirely  by  the  teachers.  In  Cleveland 
the  teachers  were  inexperienced  in  giving  tests,  while  in  Grand 
Rapids  they  were  all  more  or  less  familiar  with  the  Courtis  tests. 

GENERAL  RESULTS 

I.  Standard  scores  for  the  several  sets  in  Grades  3-8  have  been 
determined  on  the  basis  of  results  secured  from  Cleveland  and 

t  I2Z 


122        STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

Grand  Rapids  pupils.  A  comparison  of  these  scores  with  the 
Courtis  standard  scores  in  Sets  A,  B,  C,  and  D  indicates  that  the 
scores  in  these  sets  constitute  rather  accurate  standards  of  attain- 
ment; and  there  seems  to  be  no  reason  for  believing  that  the  scores 
in  the  other  sets,  with  the  possible  exception  of  Set  H,  do  not  con- 
stitute equally  accurate  standards. 

2.  A  system  of  weights  has  been  derived  whereby  it  is  possible 
to  equate  the  scores  made  in  the  several  sets  by  an  individual  or 
group  so  that  a  single  score  may  be  secured  to  represent  the  general 
arithmetical  attainment  of  the  individual  or  group. 

3.  The  use  of  the  test  is  considered  at  some  length.  Methods 
of  diagnosing  individual,  class,  school,  and  city  weaknesses  are 
indicated. 

4.  Some  very  interesting  facts  are  brought  out  in  comparing 
grade  distributions  in  the  various  types  of  examples:  first,  in  the 
fundamentals  the  distribution-curve  tends  to  become  flattened  with 
progress  through  the  grades;  second,  the  distribution-curve  also 
tends  to  become  flattened  as  we  proceed  from  the  less  complex  to 
the  more  complex  types  of  examples  in  the  fundamentals;  third, 
as  a  general  proposition  in  the  fundamentals  the  distribution-curve 
representing  the  "rights"  is  flatter  than  that  representing  the 
"attempts";  fourth,  in  set  O,  fractions,  the  exact  reverse  of  this 
last  statement  is  true,  the  curve  for  the  "attempts"  being  flatter 
than  that  for  the  "rights." 

5.  Tentative  standards  of  accuracy  for  each  of  the  sets  in 
Grades  3-8  inclusive  have  been  determined  on  the  basis  of  results 
from  Cleveland  and  Grand  Rapids  children. 

6.  Curves  representing  progress  in  accuracy  through  the  grades 
and  curves  representing  progress  in  the  average  number  of  examples 
worked  are  compared.  The  accuracy-curve  takes  the  form  of  the 
learning-curve,  while  the  "  rights  "-curve  does  not. 

TYPES   OF  ERRORS 

I.  In  the  addition  of  the  simple  combinations  the  general  propo- 
sition seems  to  be  established  that  on  the  average  those  combina- 
tions whose  sums  exceed  ten  are  more  diflScult  than  those  whose 
sums  are  less  than  ten.    To  this  general  statement  there  are  indi- 


SUMMARY  AND  CONCLUSIONS  123 

vidual  exceptions  which  indicate  the  formation  of  f>eculiarly  strong 
associations,  some  being  right  and  others  wrong.  These  peculiar 
associations  vary  among  different  groups.  This  would  indicate  that 
the  formation  of  the  association  is  to  be  accounted  for  in  terms  of 
the  experience  of  the  group  rather  than  in  the  character  of  the 
combination  itself. 

2.  In  the  simple  subtraction  combinations  "bridging  the  tens" 
is  found  to  be  a  relatively  much  more  difficult  operation  than  in 
the  addition  combinations.  Freakish  errors,  on  the  other  hand,  are 
found  to  be  less  frequent  in  the  former  than  in  the  latter.  The 
understanding  of  the  meaning  of  zero  seems  to  accompany  maturing 
of  the  pupil.  This  is  indicated  by  a  relatively  large  percentage  of 
errors  made  on  the  combination  i— o  by  fifth-grade  pupils,  whereas 
this  combination  presented  but  little  difficulty  to  pupils  in  the 
eighth  grade. 

3.  Practically  all  the  errors  made  in  the  simple  multiplication 
combinations  are  made  in  those  combinations  in  which  zero  enters 
as  one  of  the  terms.  Furthermore,  it  is  a  more  difficult  mental 
operation  to  multiply  a  quantity  by  zero  than  to  perform  the  reverse 
operation,  multiply  zero  by  the  quantity.  And  a  pupil  may  have 
difficulty  with  the  zero  in  the  simple  combinations,  yet  be  quite 
able  to  handle  it  in  the  more  complex  examples,  and  vice  versa. 
In  the  complex  multiplication  examples  the  most  frequent  error  is 
made  in  multiplying. 

4.  In  the  simple  division  combinations  the  most  frequent  error 
is  made  in  dividing  a  quantity  by  itself.  The  result  given  is  zero, 
showing  a  confusion  between  the  division  and  subtraction  processes. 
In  long  division  the  demand  for  multiplication  accounts  for  most 
of  the  errors. 

5.  The  typical  errors  made  in  working  fractions  indicate,  as  a 
general  rule,  a  slavish  adherence  to  the  mechanics  of  fractions  and 
show  emphasis  upon  method  rather  than  upon  an  understanding 
of  the  process.  There  consequently  follows  a  great  deal  of  confu- 
sion of  methods  on  the  part  of  the  pupil. 

6.  In  the  addition  and  subtraction  of  fractions  of  like  denomi- 
nator there  is  a  tendency  to  add  both  numerators  and  denominators 
in  the  one  case  and  subtract  them  in  the  other. 


124        STUDIES  IN  THE  PSYCHOLOGY  OF  ARITHMETIC 

7.  In  the  working  of  fractions  of  unlike  denominator  those 
involving  subtraction  are  found  to  be  the  most  difl&cult,  followed  in 
order  of  decreasing  difl5culty  by  those  involving  addition,  division, 
and  multiplication.  Multiplication  of  such  fractions  is  shown^to 
be  especially  easy. 

8.  In  the  application  of  each  of  the  fundamental  operations  to 
fractions  there  seem  to  be  certain  types  of  errors  which  recur  again 
and  again.  Careful  attention  on  the  part  of  the  teacher  to  these 
typical  errors  would  be  worth  while. 

A   COAIPARISON   OF   THE  ARITHMETICAL  ABILITIES   OF   CERTAIN  AGE 
AND  PROMOTION   GROUPS 

1.  With  reference  to  the  number  of  examples  worked  correctly 
by  the  pupils  in  the  four  age  groups,  it  may  be  said  that  on  the 
average  the  younger  groups  are  superior  to  the  older  groups;  that 
this  superiority  is  more  marked  in  the  later  than  in  the  earlier 
grades ;  and  that  it  is  also  more  marked  in  the  handling  of  the  more 
complex  than  in  the  handling  of  the  simpler  types  of  examples. 

2.  In  the  number  of  examples  attempted  the  study  reveals  no 
clear  difference  between  any  two  of  the  four  age  groups. 

3.  On  the  average,  the  younger  pupils  are  found  to  be  more 
accurate  in  their  work  than  the  older  pupils;  these  differences  are 
on  the  whole  quite  uniform  from  grade  to  grade;  and  they  are  more 
pronounced  in  the  more  complex  than  in  the  simpler  examples. 

4.  A  study  of  "fast,"  "regular,"  and  "slow"  pupils,  as  deter- 
mined by  promotion  facts,  reveals  differences  of  the  same  order  as 
those  just  stated  concerning  the  age  groups,  the  "fast"  correspond- 
ing to  the  "young"  and  the  "slow"  to  the  "old." 

5.  A  regrouping  of  these  same  pupils  ("fast,"  "regular,"  and 
"slow")  on  the  basis  of  age  shows  the  differences  to  be  more  pro- 
nounced than  when  grouped  according  to  the  rate  of  promotion. 

6.  This  last  statement  would  indicate  a  tendency  to  keep  pupils 
in  a  grade  because  of  youth. 

7.  "Failures"  (pupils  repeating  because  of  inability  to  do  the 
work  of  the  grade)  are  inferior  to  "irregular"  pupils  (pupils  repeat- 
ing because  of  sickness,  transfer  of  school,  etc.),  and  the  latter  are 
inferior  to  "regular"  pupils  (pupils  making  just  normal  progress). 


SUMMARY  AND  CONCLUSIONS  125 

The  relation  between  the  two  latter  groups  is  of  significance  as  indi- 
cating injurious  effects  of  repeating. 

8.  The  evidence,  inconclusive  because  of  the  small  number  of 
cases  involved,  indicates  that  among  eighth-grade  pupils  the  group 
made  up  of  pupils  who  had  failed  below  the  sixth  grade  is  superior 
to  the  group  composed  of  pupils  who  had  failed  above  the  fifth 
grade. 

A  COMPARISON  OF  THE  ARITHMETICAL  ABILITIES  OF 
CERTAIN  RACE   GROUPS 

The  conclusions  that  may  be  drawn  from  this  study  must,  of 
course,  be  more  or  less  tentative  because  of  the  small  number  of 
cases  available.  It  seems  safe,  however,  to  conclude  that  the  differ- 
ences in  arithmetical  abilities  of  children  of  American  parentage  and 
children  of  Holland  descent  are  very  small,  if  they  exist  at  all.  The 
same  may  be  said  of  the  German  children,  as  compared  with  these 
two  groups,  while  the  indications  are  that  the  Swedes  and  the 
Slavs,  and  especially  the  latter,  are  superior.  But  as  to  whether 
these  differences  are  due  to  the  operation  of  biological  or  social 
factors  the  present  study  furnishes  no  evidence. 


INDEX 


Accuracy,  47;   curves  of,  50. 

Addition,  6;  standards  in,  22;  distribu- 
tions of  pupils  in,  38;  accuracy  in,  48; 
t5^es  of  error  made  in,  53;  of  fractions, 
64,  69. 

Age  groups:  arithmetical  abilities  of,  78, 
105,  124;  method  of  determining,  78; 
results,  81;  rights,  81;  attempts,  94; 
accuracy,  96;   conclusions,  no. 

Americans,  112. 

Arithmetical  abilities:  of  age  groups,  78, 
105;  of  promotion  groups,  78,  loi; 
of  race  groups,  112. 

City  systems,  comparison  of,  28. 

Class  weaknesses,  32. 

Classes  tested,  19. 

Cleveland  test,  i,  3,  18,  21,  28. 

Collection  of  data,  18. 

Comparison  of  city  systems,  28. 

Courtis  standards,  23. 

Courtis  tests,  i,  23;  series  A  and  B  of,  3, 

4,  23. 
Curves  of  accuracy,  50. 

Data,  collection  of,  18. 

Derivation  of  system  of  weights,  26. 

Determination  of  standards,  21. 

Diagnosing:  class  weaknesses,  32;  indi- 
vidual weaknesses,  35. 

Distributions,  36;  in  addition,  38;  in 
multiplication,  42;   in  fractions,  43. 

Division,  16;  standards  in,  22;  accuracy 
in,  48;  types  of  error  made  in,  62;  of 
fractions,  69,  75. 

Failures,  102,  106. 

Fast  pupils,  loi,  103. 

Fractions,  17;  standards  in,  22;  distri- 
butions of  pupils  in,  43;  accuracy  in, 
48;  types  of  error  made  in,  64;  of  like 
denominators,  64;  of  unlike  denomina- 
tors, 68. 


General  results,  21,  121. 

Germans,  112. 

Grand  Rapids  test,  i,  19,  21,  28. 

Hollanders,  112. 

Individual  weaknesses,  35. 
Introductory  statement,  i. 
Irregular  pupils,  102,  106. 

Multiplication,  16;  standards  in,  22; 
accuracy  in,  48;  types  of  error  made  in, 
58;  of  fractions,  69,  72. 

Number  of  classes  tested,  19. 

Promotion  groups:  arithmetical  abilities 
of,  78,  loi,  124;  method,  103;  results, 
no. 

Pupils  failing  below  sixth  grade,  109; 
failing  above  fifth  grade,  109. 

Race  groups:  arithmetical  abilities  of, 
112,  125;  method  of  determining,  112; 
resiilts,  117;  conclusions,  120. 

Regular  pupils,  loi,  102,  103,  106. 

Results,  general,  21,  121. 

Series  A  and  B  of  Courtis  test,  3,  4,  23. 

Slavs,  112. 

Slow  pupils,  loi,  103. 

Spiral  character  of  test,  6. 

Standards:  determination  of,  21; 
Courtis,  23. 

Subtraction,  15;  standards  in,  22; 
accuracy  in,  48;  types  of  error  made 
in,  56;  of  fractions,  67,  69,  71. 

Swedes,  112. 

Time  allowance,  17. 

Types  of  errors,  53,  122;  in  addition,  53; 
in  subtraction,  56;  in  multiplication, 
58;  in  division,  62;  in  fractions,  64. 

Use  of  test,  28. 

Weaknesses:  class,  32;  individual,  35. 
Weights,  derivation  of  system  of,  26. 


137 


SOUTHERN  BRANCH, 

UNIVERSITY  OF  CALIFORNIA, 

LIBRARY, 

ILOS  ANGELES,  CALIF. 


2,  T  5  4      S> 


This  book  is 

DUE  on  the  last  date  sti 

imped  below 

DEC  5      193V 

APR  24:  1935 
AUG  7      1942 

.1 

SEP  2  2  19A3 

AUG  23  1952' 
APR  15  1953 

i 

J/tN'ilU-r 

/JUII12^^^ 

JAN  3    1S6^ 

Form  L-9-15m-7,'31 

Cb3 


IJJWVHRMTY  of  CALIFOKAi^ 

AT 

LOb  ANGELKS 

UBRAKY 


